Analysis the torsional vibration of PC Box Girder Bridge with Corrugated Steel Webs

. In this study, in order to accurately calculate the torsional vibration frequency of a simply supported corrugated steel web composite box girder, the element stiffness matrix of the torsional vibration of the simply supported corrugated steel web composite box girder was derived using the principle of constant dynamic potential energy, taking into account the shear deformation of the corrugated steel web. Analyzed the influence of waveform steel web plate type and shear deformation on torsional vibration frequency. The results show that the torsional vibration frequencies of the four commonly used types of simply supported corrugated steel web composite box girders in engineering are relatively close, and the influence of corrugated steel web types on torsional vibration frequencies can be ignored in practical calculations; The influence of shear deformation on the torsional vibration frequency of simply supported corrugated steel web composite box girder is significant, and exceeds the influence on its bending vibration frequency. Therefore, the influence of shear deformation must be considered in practical engineering calculations


Introduction
The corrugated steel web PC box girder bridge is a new type of steel-concrete composite bridge structure.Its obvious feature is to replace the 30-80mm thick concrete web with a corrugated steel web of about 10mm thickness [1] , and apply prestressing to the beam body through internal and external prestressing tendons.This bridge type significantly reduces the self weight of the beam and improves the efficiency of prestressed application of the beam.Due to the special structure of corrugated steel web plates, their shear strength is significantly improved compared to flat web steel plates.Due to its many advantages, this bridge type has been increasingly used in bridge construction both domestically and internationally.However, due to the use of thinner corrugated steel plates as the web plates of the box girder, its torsional stiffness is significantly weakened compared to traditional prestressed concrete box girder bridges [2] .
At present, domestic and foreign scholars mainly focus on the dynamic performance research of simply supported corrugated steel web composite box beams in terms of bending and shear performance.Hassanein et al. [3] studied the shear buckling performance of I-shaped steel beams with corrugated steel web plates.Oh et al. [4] studied the influence of accordion effect on the mechanical properties of corrugated steel belly plates.Jun et al. [5] derived a formula for calculating the flexural strength of corrugated steel web composite I-beams under symmetric loads; K ö vesdi et al. [6] studied the mechanical behavior of I-shaped steel beams with corrugated steel web plates under the combined action of bending moment and shear force; Elamary et al. [7] studied the bending and bearing capacity performance of corrugated steel web steel-concrete composite box girder bridges; Lee et al. [8] studied the effects of the arrangement of external prestressed tendons, welding methods, and shear connectors on the structural performance of I-shaped composite beams with corrugated steel web plates.On the basis of considering shear deformation, Luo Kui et al. [9] developed a deflection calculation program for a corrugated steel web composite box girder bridge using MATLAB software.In recent years, some scholars have also conducted relevant research on the torsional performance of simply supported corrugated steel web composite box girders.Reference [10] derived a formula for calculating the torsional vibration frequency of a simply supported PC box girder bridge with corrugated steel web plates; Ren Hongwei et al. [11] derived a theoretical calculation method for the torsional vibration of concrete box girders with corrugated steel web plates, taking into account the influence of transverse diaphragms on torsional vibration; Nie Jianguo et al. [12] studied the influence of shear deformation of corrugated steel web on the mechanical behavior of corrugated steel web composite beams.
Based on the above analysis, the research on the torsional vibration characteristics of corrugated steel web PC composite box girder bridges is still relatively lagging behind [13] .In order to accurately calculate the torsional vibration frequency of PC box girder bridges with corrugated steel webs, based on the study of shear deformation of corrugated steel webs, the author applied the principle of constant dynamic potential energy and the "matching seat" rule to obtain the calculation formula for the torsional vibration frequency of PC simply supported box girder bridges with corrugated steel webs, and derived the element stiffness matrix and element mass matrix for the torsional vibration of simply supported corrugated steel web composite box girders.On the basis of the derived matrix, a program for solving the torsional vibration frequency of a simply supported corrugated steel web composite box girder was developed using MATLAB programming software.The research results can simplify the calculation process of the torsional vibration frequency of this bridge type and provide a certain reference basis for the calculation and research of the torsional vibration frequency of this bridge type.

Theoretical derivation of box girder vibration 2.1. Calculation Assumption
The vibration theory of box girder indicates that the vibration of box girder is usually accompanied by three forms: torsional vibration, horizontal flexural vibration, and vertical flexural vibration, and these three vibrations are coupled with each other.In practical calculations, the influence of coupling terms can be ignored, and the torsional vibration of box girder can be regarded as only the torsional free vibration of box girder.During this process, only rigid torsion will occur.When a simply supported box beam undergoes torsional vibration, the longitudinal warping of its cross-section is constrained, and the torsion that causes free warping at each point on the contour line of the member is called free torsion or Saint Venant torsion.If the longitudinal warping of its cross-section is constrained (or obstructed), this type of torsion is called constrained torsion.,The points on the contour line of the member not only generate relative displacement in the plane, but also warp out of the plane.Therefore, during the process of torsional free vibration, the structure is not affected by the load, so its rigid torsion remains in a free torsion state.
1).Neglecting the influence of steel reinforcement on the free vibration of beams; 2) When simply supported corrugated steel web composite box girder is twisted, only the free torsion around the shear center occurs; 3).The roof to web connection is assumed to be rigid; 4).When the beam body is under torsion, the longitudinal fiber of each section will not be stretched and warped, that is, the section of the beam body will only produce the free torsional shear stress, but not the longitudinal stress.
The simply supported box beam shown in Figure 1 has a boundary condition where one end is a fixed hinge support and the other end is a movable hinge support.When the beam body is twisted, The crosssectional view of the corrugated steel web PC composite box girder is shown in Figure 2 the displacement mode is shown in Figure 3. Since the connection between the top plate and the web is assumed to be rigid connection, the dynamic displacement u and the torsion angle  can be expressed as Where 1 H is the distance from the shear center to the top edge of the tip.
Based on the theory of Timoshenko beam and considering the shear deformation of corrugated steel web, the micro-element deformation diagram of simply supported corrugated steel web composite box beam is shown in Fig. 4. The relationship between shear angle  , bending angle  and dynamic displacement u can be expressed as: The force diagram of the microelement is shown in Figure 5.

Element displacement mode
The box girder is discretized into several beam elements.The schematic diagram of any beam element is shown in Fig. 6.The displacement of any point on the middle section of the element is the superposition of the displacement caused by the dynamic displacement Since only the torsional vibration of the box girder is calculated, only the first derivative of the interpolation function is required to be continuous when selecting the interpolation function, so the element displacement interpolation function can be taken as the cubic polynomial In the formula Among them, l is the length of the beam end element

Among them, [ ( )]
N z is the shape function matrix,   are the bending angles and torsion displacement arrays at both ends of the element.

Derivation of stiffness matrix for simply supported corrugated steel web composite box girder elements
The box girder is divided into units along the span direction.According to the principle of constant dynamic potential energy, the strain energy variation of each element can be used to obtain the stiffness matrix of the element.
The unit bending strain energy is: The torsional strain energy of the unit is: The unit shear strain energy is: In the formula, 1 U , 2 U , and 3 U represent the bending, torsion, and shear strain energies of the unit, respectively.EI is the bending stiffness, GJ is the torsional stiffness, G is the shear modulus, 1 A is the shear area, and k is the shear shape coefficient.The total strain energy of the unit is: ) Substitute equations (7) to (9) into equation (10) to obtain the total strain energy of the unit as: Substituting equations ( 1) and (2) into equation ( 11) yields: F are the element stiffness matrix, node displacement matrix, and node force matrix, respectively.
In the formula, "*" represents the first-order variation of the displacement parameter, and   e *  represents the first-order variation array of the displacement parameter of the element node, 4), (5), and (14) into equation ( 13) to obtain the specific form of the element stiffness matrix:

Mass matrix of simply supported corrugated steel web composite box girder elements
According to the relevant theory of finite element analysis, the concentrated mass matrix can be used to solve the natural frequency the beam, that is, the element mass matrix of the torsional vibration of a simply supported corrugated steel web composite box beam is: In the formula:  is the mass density of the beam segment element; A is the cross-sectional area of the beam segment unit; l is the length of the beam segment unit.

Solution to torsional vibration frequency of simply supported corrugated steel web composite box girder
According to the relevant theories of structural dynamics, if it is assumed that the free vibration of the system is a harmonic vibration, the characteristic equation of the free vibration motion equation of an undamped system is: The condition for equation ( 17) to have a non-zero solution is: Equation ( 18) is called the frequency equation of the system.Based on this frequency equation, combined with the derived element stiffness matrix   e K and element mass matrix   e M of the simply supported corrugated steel web composite box girder, a program TFBOX was developed using MATLAB software to solve the torsional vibration frequency of the simply supported corrugated steel web composite box girder.Due to space limitations, specific programs will not be repeated.
Related studies have shown that the shear stress of a simply supported corrugated steel web composite box girder is uniformly distributed along the beam height direction, and the shear stress only exists within the corrugated steel web [9] .Therefore, the shear coefficient in this paper is taken as 1.0.
Due to the wrinkling effect of the corrugated steel web, the calculation formula for its effective shear modulus is as follows: In the formula, s E and s  are respectively the shear modulus and Poisson's ratio of the steel, and the other parameters are shown in Figure 7.

Example verification
To verify the correctness of the program proposed by the author for solving the torsional vibration frequency of simply supported corrugated steel web composite box girders, the calculated values of the TFBOX program in this paper were compared with the experimental values and finite element calculations in references [8] to [10].

Calculation example of simply supported corrugated steel web composite box girder
Compare the calculated values of the TFBOX program in this article with the experimental values and finite element calculations conducted in reference [8].The experimental beam is a scaled model of the already built Juancheng Yellow River Highway Bridge, with a cross-section selected from the mid span design section of the bridge, as shown in Figure 8.The test beam has a total length of 6m, with one end hinged and one end fixed.The concrete material is C50 concrete.The corrugated steel belly plate adopts Q235 steel plate, with elastic modulus and shear modulus of 206GPa and 0.3, respectively.Its cross-sectional dimensions are shown in Figure 9  The natural frequency values measured in the dynamic characteristics test of the test beam and the frequency values calculated by ANSYS finite element software were compared and analyzed with the torsional vibration frequency calculated by the MATLAB program proposed in this paper.The results are shown in Table 1.From Table 1, it can be seen that the calculated values of the MATLAB solution program for the torsional vibration frequency of the simply supported corrugated steel web composite box girder proposed in this article are in good agreement with the measured values and finite element calculation values of the experimental beam in reference [8] , verifying the correctness of the TFBOX program proposed in this article.This can provide a simple and reliable method for calculating the torsional vibration frequency of this bridge type in practical engineering.

Comparative analysis of calculation results
In order to compare and analyze the influence of waveform steel web type and shear deformation on the torsional vibration frequency of simply supported waveform steel web composite box beams, based on the waveform steel web test beam in Example, and with other parameters such as the thickness of reinforced concrete at the top and bottom plates and the thickness of the waveform steel web remaining unchanged, The torsional vibration frequencies of four commonly used waveform steel web plates in engineering were calculated with and without considering shear deformation.The dimensions of the four waveform steel web plates are shown in Table 2, and the meanings of the parameters in the table are shown in Figure 5.The frequency values calculated using the TFBOX program in this article are shown in Table 3.Among them: 1 f is the measured value; 2 f is the theoretical value considering shear deformation; 3 f is the theoretical value without considering shear deformation; From Table 3, it can be seen that the type of corrugated steel web has a relatively small effect on the frequency of torsional vibration of simply supported corrugated steel web composite box girders, which can be ignored in actual calculations.The influence of shear deformation on the vertical bending vibration frequency of simply supported corrugated steel web composite box girder is , 01016 (2024) E3S Web of Conferences ARFEE 2023 https://doi.org/10.1051/e3sconf/202449001016490 significant.After calculation, shear deformation has a high impact on the torsional vibration frequency of 41.11%.Reference [12] explores the influence of shear deformation on the vertical bending vibration frequency of simply supported corrugated steel web composite box girder.The research results show that the influence of shear deformation on the vertical bending vibration frequency of this bridge type is about 16%, This indicates that shear deformation has a significant impact on the natural frequency of simply supported corrugated steel web composite box girders, and its impact on torsional vibration frequency is greater than that on vertical bending vibration.

Conclusions
(1) The MATLAB solution program for the torsional vibration frequency of corrugated steel web plates was derived using the principle of constant dynamic potential energy and the "matching seat" rule to form matrices.The calculated values of the program were compared with the measured values and finite element values, verifying the correctness of the program TFBOX proposed in this paper.This provides a reliable and convenient calculation method for the torsional vibration frequency of simply supported corrugated steel web plate composite box beams in practical engineering; (2) The calculation formulas for shear coefficient, effective shear modulus, and torsional stiffness proposed in this article can be used to calculate the cross-sectional characteristics of this bridge type; (3) The torsional vibration frequency of simply supported corrugated steel web composite box girder is less affected by the type of corrugated steel web, so it can be ignored in practical engineering applications;

Fig. 1
Fig. 1 Boundary Conditions of Simply Supported Beam.

Fig. 5
Fig. 5 Force Acting on the Element.

Substitute equation ( 3 )
into the above equation, and derive the form of element stiffness matrix e K ] [ from the variation of element strain energy as follows:The stiffness matrix of a beam element can be expressed as

Table 1 .
Comparison of frequencies for the test girder(unit:Hz)

Table 2 .
Dimensions of corrugated steel webs of standard shape