Dynamic Analysis of Curved Box Girder Bridges

(0.5, 0.25) of a curved box girder bridge with simple support and exposed to the traffic of the moving vehicles represented by the system of the moving force. Where the dynamic interaction between the vehicle and the bridge is neglected. The study has also required the application of Newmark’s method to find out the dynamic response values represented by (deflections, dynamic amplification factor, Impact factor) to solve the issues related to the bridge loaded by moving vehicles model. Each system is composed of a suspended mass joined with a damper and a spring of stiffness in addition to the unsuspended mass.


Introduction
Vibration of bridge structure under the passage of vehicles is an important consideration in the design of bridges. Dynamic response of bridges has assumed added significance with the advent of faster and heavier vehicles and the use of structural forms and materials that permits the bridge to be more slender. Interaction between the vehicle and the bridge is complex dynamic phenomenon. This complexity results from the large number of different parameters that may affect the dynamic response [1].
During the past decades, extensive works have been under taken to study the factors affecting the dynamic response of bridges, Mukherjee and Trikha [2] investigation the influence of span, radius of curvature, span/depth ratio and the speed of moving load have been indicated on the maximum values of dynamic factors for individual girders.
In 1997, Senthilvasan etal. [3] presented a simple procedure for modeling and analyzing the dynamic behavior of curved box girder bridges based on the curved spline finite strip method, investigation the influence of several important parameters such as: (1) The curvature of bridge, (2)mass ratio (mass of vehicle/ mass of the bridge), (3) speed of vehicle, (4) support conditions, etc.
To investigate the dynamic response of box -girder bridge Abdul-Razzak and Mohammad in (2005) [4] study a higher order finite strip with sixth order bending strips combined with third order in-plane displacements functions have been used to obtain the analysis of simply supported box-girder bridges under moving force.
In (2007) Abdul-Razzak and Haido [5] studied the effects of some parameters on the vibration of the plates subjected to dynamic load. The forced vibration analysis by using a higher order finite layer formulation based on the auxiliary nodal surface (A.N.S.) technique for analysis of rectangular plates, by applying the Newmark method for investigating the vibration characteristics and finding the response of the rectangular plates under the action of dynamic loads In (2008) Abdul-Razzak and Haido [6] investigated the dynamic response of beam under moving vehicles. The bridge is idealized by a rectangular isotropic or orthotropic plate and vehicle is represented by a one-foot dynamic system with the unsprung mass and sprung mass interconnected by a spring and a dashpot. Higher order finite layer method have been used to simulate the dynamic interaction between abridge deck and moving vehicle.
In present study, a refined finite strip method is applied for the force vibration of curved box-girder bridges using higher order conical frustum shell strip. The auxiliary nodal line (A.N.L.) technique has been adopted for both bending and membrane actions. The vehicle has been represented by single moving force with the same velocity as the vehicle.

Formulation of Shell Strip Characteristics: Basic Assumption
For the general conical shell strip with one auxiliary nodal line, a sixth order displacement function to represent the variation of the normal displacement components W was used, and a third order displacement function to represent the variation of the membrane displacements U and V were used, as follows [7]: where: 6

Strain
The equations of strain displacement relations for a conical shell have been derived by Novozhilov [8] , and they are found to be most appropriate. (12) in simple form:

Stresses
The stresses encountered in the strain energy of shell strip, for the orthogonal anisotropic material, are given by [7] :  (15) in which :

Stiffness and Force Matrices
The total potential energy of a conical shell strip subjected to a load q can be expressed as [7,9] :

Modeling of the vehicle and bridges
The use of curved bridges in interchanges of modern highway systems is popular because of increased demand for curved roadway alignments for the smooth passage of congested traffic and modern emphasis on aesthetic considerations. Box girders are the most preferred section for curved bridges on account of their high torsional capacity [10].
The vehicle is idealized as a single moving force by neglected the dynamic interaction between the vehicle and bridge, also the vehicle is idealized as a moving mass consist of (sprung mass-damped-spring) with dynamic interaction is considered.

Equations of the dynamic response
The forced vibration equation defining motion may be expressed as [11,12,13]: and its derivatives at the instant (t+Δt) can be related to those at the instant [14].
where the quantities with subscript t are those occurring at time t, assumed to be known. Using New mark's parameters β and γ, the coefficient and those to be used later can be given as: And forming the effective stiffness matrix   * K : It will be possible , for each time step , to calculate the dynamic response in the following steps: 1.calculate the effective force vector at time t+ Δt as following The equation of motion for sprung mass express by the following matrix [13,15,16].
Calculate the effective force for sprung mass at time t+ Δt as following:

Numerical Example:
Dynamic analysis of curved box-girder bridges with two cell (Bridge-1).
The model, which was tested in ref. [17], was a two cell curved box girder bridge under (345 kN) moving force, the cross section dimensions are shown in Fig. (1)  The transverse section is discredited using (11) higher order finite strips as shown in Fig. (2), the speed of vehicle is (110 km/hr) along the longitudinal centerline of the bridge. The dynamic increment factor (Imp.), for the center line of bridge is given in Fig(7)., results are in good agreement with the values obtained by the finite element method [17].

Conclusions:
Sixth order finite strip and third order in plane strip with auxiliary nodal line are used successfully for the solution of curved box-girder bridges under dynamic response.
The normalized displacement increased by increasing the distance of the moving force on the bridge until it reach a center of space after that is begin by decreasing.
Comparison of the results shows that an increase in the dynamic deflcetion compared to static deflection. The dynamic deflection is approximately (26.5%) greater than the maximum static deflection.
Accurate results are found using the higher order finite strip method when compared the finite element solution.