Abstract
A onedimensional highorder dynamic model for singlebox twincell box girders is presented together with the pattern recognition algorithm. The model takes into account the deformable crosssection and can accurately predict its 3D dynamic behaviors. The crosssection deformation is captured by basis functions satisfying displacement continuity condition, which is essential to construct the initial model formulation based on the Hamilton principle. The axial variation patterns of generalized coordinates are decoupled by solving the eigenvalue problem. On this basis, the combinations of basis functions are obtained to bring out crosssection deformation. The crosssection deformation, hierarchically organized and physically meaningful, are used to update the basis functions in the reconstructed highorder model. Numerical analysis has verified the accuracy and applicability of the reconstructed onedimensional highorder model.
Highlights
 A onedimensional highorder dynamic model for singlebox twincell box girders is presented together with the pattern recognition algorithm. A set of shape functions is defined to capture crosssectional deformation to construct the onedimensional highorder initial model.
 The crosssection deformation is captured by principal component analysis and used to reconstruct highorder model in order to reduce the degree of freedom of the model and improve computational efficiency.
 The reconstructed highorder model, which is compared with the ANSYS model, is validated for the applicability of threedimensional dynamics, slenderness ratio and highorder deformation.
1. Introduction
With the continuous development of transportation and engineering technology, bridge structures have become widely used in highways. Box girders play an important role in the loadbearing capacity of bridge structures, due to their excellent seismic resistance, bending and torsion resistance, and overall loadbearing performance. As research progressed the fabrication of the box girders have undergone several variations from the initial use of castinplace brackets to later prefabricated installations. There have also been some changes in the crosssection configuration of box girders, such as the number of box chambers and the form of composite box sections. The traditional singlebox singlecell section structure [1][3] is lightweight and has strong applicability, but the singlebox twincell box girder bridge gradually appears in the public's view with the rapid increase of traffic flow [4][5]. The stress characteristics of the box girder are completely different from those of the singlebox singlecell box girder. At the same time, the rapidly growing traffic demands emphasize the need for box girder bridges with more and more lanes. The singlebox singlecell box girder bridge cannot meet the requirement. Therefore, it is inevitable to research and develop twincell box girder bridges.
Box girders generally have a large span and a crosssection constructed with thin walls, belonging to typical thinwalled structures. The theories of thinwalled structures have been developing constantly, which can provide supports for the mechanical analysis of box girders. Usually, various external forces borne by the box girder section are divided into symmetrical dead loads and symmetrical or asymmetric live loads. The box girder will deform for bending and torsion under these loads. To rationally evaluate its performance, researchers have made quantities of efforts. The measures adopted are generally divided into model experiment method and analytical method. The former method usually reduces the box girder to a certain proportion to study its characteristics. For example, Chithra et al. [6] studied the formation process of cracks on the surface of the twincell box girder bridge under the combined action of bending and torsion, and Androus et al. [7] investigated free vibration and ultimate behavior of composite twinbox girder bridges to study the effect of internal cross bracing between box girders along with the curvature effect at different loading stages.
Analytical methods have played important role in explaining special behaviors of box girders. Typically, some researchers have considered the effect of initial curvature to obtain analytical expressions which describes the bendingtorsion coupling characteristics of curved box girders under different loads. They provide a theoretical basis for the application under special terrain conditions [2][4]. Taking the shear of box girder into account, Zhu et al. [8] developed a modeling strategy to describe the distortion of composite trapezoidal box girders with cantilever overhangs while considering various indices and shear deformation of inner crossframes. Zhang et al. [9] provided a solution to improve the buckling and shear performance for the bionic box girder bridge with steel webs. At the same time, the phenomenon of shear lag in the wing plates of wide flange box girders has a notable impact on structural performance of box girders. Some scholars have revealed the distribution of shear flow in box girders [10][12].
The stress state of box girder bridges under various loads during construction and operation stages is more complex. In addition to bending and torsional deformation, spatial effects such as distortion and local lateral bending are prominent. Classical beam theory, such as Euler beam [13], is a simplified form of linear elastic theory for beams and is widely used for loadbearing and deformation calculations. However, it ignores the effects of section warping and distortion. Currently, in order to accurately characterize its 3D structural behavior, the highorder beam theory [14] has been further developed by considering crosssectional characteristic deformation. On the basis of traditional beam theory, the deformation mode of special behavior is considered to enhance the capability to predict crosssection deformations. Vlasov theory [15][16] is among the earliest ones in analyzing highorder beams. It defines crosssectional warping and includes crosssectional deformation shapes as additional degrees of freedom.
Later, the concept of generalized warping function was further used by many authors. For example, R. Vieira [14] provided a procedure to define uncoupled warping modes within the framework of a higherorder beam model. Choi et al. [16] proposed a higherorder beam bending theory, which not only includes as many bending related section modes as possible, but also provides the required explicit relationship between stress and generalized forces. To evaluate the vibration of thinwalled beams with arbitrary open sections, Jrad et al. [17] derived an analytical expression for the beam in bendingtorsion coupled modes. Kim et al. [18] derived the crosssectional shape functions of highorder deformation modes, which are used to analyze thinwalled beams composed of straight section edges. Zhou et al. [19] put forward the analytical deflection theory considering the strain and displacement of each plate in a singlecell box girder. The shear deformation of box girder web and flange is considered under continuous deformation condition. Vieira et al. [20] proposed a procedure to calculate crosssection deformation modes of steelconcrete composite bridge. Henriques et al. [21] presented and validated a finite element that combines the effects of concrete creep and crosssection deformation, namely distortion and shear lag. Barrientos et al. [22] proposed a onedimensional element for thinwalled beams considering torsion, deformation and shear hysteresis.
In this paper, a onedimensional highorder dynamic model is presented which considers the crosssectional deformation of a singlebox twincell box girder. First, a set of basic deformation modes of the crosssection is defined, and shape functions are used to capture the displacement of the crosssection. The initial onedimensional highorder dynamic model is presented in Section 2. The principal component analysis method is employed to efficiently extract crosssectional feature deformation patterns with certain physical significance, and the dynamic model is further updated using feature deformations in Section 3. In the identification procedure, geometric and boundary conditions are also considered. In Section 4, the universality and effectiveness of the updated onedimensional highorder dynamic model are verified through comparison with shell model. Finally, the main conclusions are outlined.
2. Onedimensional highorder beam model
2.1. Displacement field
The research object of this paper is the twincell box girder with equal crosssection shown in Fig. 1(a). The parameters $L$, $h$, $b$, $a$, and $t$ respectively represent the overall length, web height, bottom plate width, wing plate length and wall thickness of the box girder. To describe the displacement field, a global coordinate system $\left(x,y,z\right)$ is set with its origin located in the centroid of the crosssection at one end of the box grider. The $z$axis is parallel to the beam axis, while the $x$axis and $y$axis represent the tangential and normal directions of the crosssection, respectively. Thinwalled sections are discretized along the centerline, and the local coordinate system $\left(s,n,z\right)$ is adopted on the wing plate and web of the twincell box girder in Fig. 1(b), where the $s$axis and $n$axis define the centerline plane of the plate, and $z$axis represents the direction along the axis.
Fig. 1The singlebox twincell girder: a) the global coordinate system, b) the local coordinate system, and c) discretization nodes of the crosssection
a)
b)
c)
The displacement field is built to accurately describe the crosssection deformation in the means of combinations of a set of shape functions defined on the crosssection. These shape functions are onedimensional and satisfy displacement continuity conditions along the centerline.
As shown in Fig. 1(c), the twincell box girder crosssection can be divided into a set of walls, connected by eight natural nodes, which connects adjacent walls or locates at the free ends. Since three webs may be relatively large in length, appropriate node refinement will contribute to the capability of capturing crosssection deformation from the viewpoint of interpolation [23]. Besides, three artificial nodes are addtionally employed to discretize the centerline in Fig. 1(c). Unit displacements are applied to each node along the axial, tangential, normal, and rotation direction, as shown in Fig. 2. The adjacent nodes are constrained to have zero displacement. As a result, the discrete nodes on the twincell box girder generated four basic deformation modes, including one outofplane deformation mode and three inplane deformation modes. Therefore, a total of fortyfour basis functions are defined on the discrete crosssection, enabling a clear expression of fortyfour basic deformations.
According to onedimensional highorder theory, the displacement field $\mathbf{h}$ of the box girder section is described by three components, namely $u\left(s,z\right)$, $v\left(s,z\right)$, and $w\left(s,z\right)$, respectively. They are approximated by a set of basis functions about $s$, which are formulated by the displacement interpolation between crosssection nodes. The displacement field $\mathbf{h}$ are specifically represented as:
where ${\mathbf{a}}_{i}\left(s\right)$ is the basis function vector using one time Lagrange interpolation, representing the axial deformation outside the section.${\mathbf{\psi}}_{i}\left(s\right)$ and ${\mathbf{\beta}}_{i}\left(s\right)$ are the basis function vectors of tangent $s$ and normal $n$ deformation in the plane, respectively, which are obtained by Lagrange and Hermite interpolation functions:
Fig. 2Basic deformation modes of the twincell box girder crosssection with eleven discrete nodes
The discrete nodes on the centerline of the crosssection of the twincell box girder generate fortyfour basic deformations. They can accurately express the displacement changes of the twincell box girder section. $\mathbf{\gamma}\left(z\right)$ is introduced to represent the weight matrix corresponding to the fortyfour basis functions, and can be expressed as:
According to Kirchhoff’s thinplate assumption, the displacement field $\mathbf{H}$ of the twincell box girder is represented by three components, namely $U\left(s,n,z\right)$, $V\left(s,n,z\right)$ and $W\left(s,n,z\right)$ for the axial, tangential and normal components, respectively:
Based on the basic assumption of Saint Venant Kirchhoff elastic body, each box girder element exhibits linear, elastic, and isotropic behavior. The strain vector ε of the twincell box girder is obtained as:
where ${\mathbf{C}}_{2}$ is a differential operator. According to the linear elastic Constitutive equation, the stress vector $\mathbf{\sigma}$ of the twincell box girder is derived from the generalized Hooke’s law:
where, $\mathbf{R}$ is the constitutive matrix, $E$ describes the elastic modulus, $G$ denotes the shear elastic modulus, and v represents the Poisson’s ratio of material.
2.2. Governing equations
The governing equation of the twincell box girder is derived by using the Hamiltonian principle. It reads:
where, ${t}_{1}$ and ${t}_{2}$ are boundary time, $T$, $U$, and $W$ are the kinetic energy, strain energy of the twincell box girder and the external force work acting on the twincell box girder. Each component can be expressed as:
where $\rho $ is the material density, $\mathbf{F}$ is the column vectors of distributed forces acting on the crosssection [2, 25], $V$ and $A$ are the volume and crosssectional area of the twincell box girder. By substituting Eqs. (810) into Eq. (7), the dynamics equation of the box grider can be derived as follows:
For ease of calculation, the finite element method is used to discretize the box grider into $m$ elements along the axial direction:
where $\mathbf{N}$ is the interpolation function in the axial direction and ${\mathbf{d}}_{i}$ is the generalized displacement vector. The vector $\mathbf{N}$ and ${\mathbf{d}}_{i}$ are expressed as follows:
where ${\zeta}_{1}$ and ${\zeta}_{2}$ are linear interpolation functions and the subscripts ($i$) and ($i+1$) indicate the ends of an element.
The singlebox twincell box girder in this paper is in a free vibration state without damping, so the load vector $\mathbf{F}$ is treated as zero. Substituting Eqs. (1214) into Eq. (11) leads to the following equation:
The form of the dynamics equation Eq. (15) can be rearranged as:
where $\mathbf{m}$ is element mass matrix, $\mathbf{k}$ is element stiffness matrix and $\mathbf{f}$ is element load matrix. The expressions of $\mathbf{m}$ and $\mathbf{k}$ are as follows:
The governing equation Eq. (11) can be used for dynamic modeling of the twincell box girder. However, the displacement field established based on the concept of Kinematics uses a large number of basis functions, which increases the calculation cost. In order to further reduce the degrees of freedom of the model, it is necessary to find a compact set of shape functions to replace these fortyfour basis functions. A set of shape functions, representing crosssectional deformations with a welldefined physical meaning, can be reconstructed for a refined highorder girder model. This will significantly improve computational efficiency while ensuring accuracy.
3. Pattern recognition
3.1. Data processing
The generalized eigenvalue matrix and generalized eigenvector matrix are obtained by solving the eigenvalue problem related to Eq. (11) using the finite element method. Data preprocessing is necessary in order to obtain the crosssectional feature deformation of thinwalled structures. The modal vectors in the $n$ modes except the first six modes are selected to form a modal vector matrix ${Q}_{1}$. It can be expressed as:
where ${q}_{n}$ is the $n$th generalized eigenvector.
According to the priority relationship, the vector ${q}_{n}$ is divided by the corresponding natural frequency of each mode. It is then sorted into a new eigenmatrix, which is further normalized to create a new modal vector matrix ${Q}_{2}$:
where, ${f}^{n}$ is the $n$th mode natural frequency of the twincell box girder, $m$ is the number of sections scattered along the axis of the twincell box girder, ${\mathbf{\varsigma}}^{1}$ is the modal vector of outofplane feature deformation, and ${\mathbf{\xi}}^{1}$ is the modal vector of inplane feature deformation.
The next goal is to eliminate the interference of the deformation patterns of the first six modes to the subsequent identification of the deformation. The classical feature deformation vector generated by the first six rigid body modes and the deformation vector of each section in each mode are orthogonalized respectively. The outofplane deformation vector matrix ${\mathbf{\varsigma}}^{2}$and inplane deformation vector matrix ${\mathbf{\xi}}^{2}$ are extracted from the updated feature deformation vector matrix $\mathbf{K}$, respectively:
$\left.{p}_{(n\times m)}\sum _{i=1}^{6}\left({c}_{i}\times dot\frac{\left({x}_{i},{c}_{i}\right)}{\left({c}_{i},{c}_{i}\right)}\right)\right]=\left[\begin{array}{l}{\mathbf{\varsigma}}_{11\times (n\times m)}^{2}\\ {\mathbf{\xi}}_{33\times (n\times m)}^{2}\end{array}\right]$
where, $p$ is the characteristic deformation vector of the section in the modal vector matrix ${Q}_{2}$, ${c}_{i}$ is the classical deformation eigenvector in the first six modes, and the ${x}_{i}$ meets the following conditions:
3.2. Principal component analysis
Principal component analysis is performed on the outofplane deformation vector matrix ${\mathbf{\varsigma}}^{2}$ and the inplane deformation vector matrix ${\mathbf{\xi}}^{2}$, seperately. By decentralizing the deformation vector matrix, it can be expresssed as:
Taking the eigenmatrices ${V}_{out}$ and ${V}_{in}$ into consideration, the covariance matrix is constructed to decompose and obtain eigenvalues and eigenvectors. The deformation corresponding to the maximum eigenvalue is known as the main feature deformation, while the rest is known as the secondary feature deformation.
The contribution rate of each deformation pattern is calculated to evalute its influence to the crosssection deformation. If the cumulative contribution rate is greater than the threshold, the recognition procedure will stop and the recognized eigenvectors and eigenvalues will be output. At this point, the recognized feature deformation vectors of the outofplane section, and inplane section can form the crosssection deformation mode of the twincell box grider. They will be used to reconstruct the highorder model to improve computational efficiency and accuracy.
3.3. Reconstructed highorder model
A set of feature deformations of the twincell box girder were obtained after pattern recognition, and then used to reconstruct the highorder model by considering their different proportion weights. The more discrete nodes are used in the central section, the more characteristic deformations are identified. Excessive deformation modes increase engineering computational costs and reduce its efficiency. To further reduce the degree of freedom of the model and simplify the initial highorder model, a new highorder model is reconstructed by selecting the section feature deformation with larger weights for engineering operations. In some engineering calculations demanding lower accuracy, six rigid body crosssection deformation modes can be selected to reproduce the classic Timoshenko beam model. In situations with high precision requirements, different quantities of outofplane and inplane highorder feature deformations can be added into the reconsturcted model. They can more accurately reproduce the mechanical properties of thinwalled structures and improve modeling accuracy.
New crosssectional deformations are added to form new shape function matrixs, ${\mathbf{\mu}}_{1}$, ${\mathbf{\mu}}_{2}$ and ${\mathbf{\mu}}_{3}$_{.} Multiple outofplane and inplane crosssectional feature deformation vectors are integrated into new vector matrices, ${\mathbf{I}}_{out}$ and ${\mathbf{I}}_{in}$. The shape function can be expressed as:
${\mathbf{\mu}}_{3}={\mathbf{I}}_{in}\times {\left[{\mathbf{\beta}}_{1}\left(s\right)\cdots {\mathbf{\beta}}_{44}\left(s\right)\right]}^{{\rm T}}.$
The new weight vector was used to describe the axial displacement changes of the reconstructed onedimensional highorder model by using the finite element method. Substituting the new weight vector $\widehat{\mathbf{\gamma}}$ into Eq. (11):
Eq. (25) is the governing equation of the socalled reconstructed onedimensional highorder model. An appropriate number of eigenvectors can be selected to construct the twincell box girder dynamics model that takes into account both computational accuracy and efficiency.
4. Comparison and validation tests
4.1. Extracting feature sections
A numerical example is conducted by using the twincell box girder with equal crosssection as the test member (shown in Fig. 1). Its span is $L=$40 m and the total height is $h=$3 m, with a bottom plate width of $b=$ 5 m, a wing plate width of $a=$ 1.5 m and a wall thickness of $t=$0.2 m. The material is C50 pure concrete, with elastic modulus, Poisson's ratio and density of $E=$34.5 GPa, $v=$0.2 and $\rho =$2500 kg/m^{3}, respectively [24][25].
Under the premise of ensuring computational accuracy and reducing model complexity, highorder feature deformations with significant weights are used to reconstruct the initial highorder model. Since the classical deformation mode accounts for a large proportion of the first six modes, this paper directly selects modal vectors of the 7th60th modes for pattern recognition. The fifteen sectional feature deformations were identified in Fig. 3. The first six modes are feature deformations in Timoshenko theory. The first three types of deformation outside the plane represent the axial extension, rotational displacement around the xaxis, and yaxis of the crosssection, respectively. The first three types of deformation in the plane represent the torsion of the crosssection and the translational displacement along the xaxis and yaxis. The nine feature deformations identified at the last represent the mechanical characteristics of the twincell box girder structure, such as warping and distortion. They represent highorder feature deformations introduced to improve computational accuracy.
Fig. 3Pattern recognition of inplane and outofplane deformation modes
For the convenience of calculation, the onedimensional highorder model and reconstructed model were both calculated using the finite element method, and eighty onedimensional highorder elements were discretized axially by linear interpolation. Both ends of the twincell box girder were fixed, and then the natural frequency of the model was calculated. At the same time, a Shell 181 element model of twincell box girder was established in ANSYS software. The first fifteen natural frequencies were calculated through modal analysis.
In order to verify the accuracy of the onedimensional highorder model considering crosssectional deformation, the natural frequencies derived with the Timoshenko beam theory, the initial model, and the reconstructed model were compared with those of the ANSYS shell model. The results are shown in Table 1. The ${f}_{1}$ is the solution of ANSYS shell element, and ${f}_{2}$ is the result obtained from classical beam theory that only considers low order deformation. The ${f}_{3}$ and ${f}_{4}$ correspond to initial onedimensional highorder models and the reconstructed onedimensional highorder models herein, and ${u}_{1}$, ${u}_{2}$, and ${u}_{3}$ represent the relative errors calculated based on the ANSYS shell model, respectively.
The relative error between Timoshenko beam theory and ANSYS shell element is relatively large after the second mode. Although it improves computational efficiency, the accuracy of the results is very low. The results of the initial highorder model is relatively more accurate, with relative error less than 2.42 % compared to the shell element.
The natural frequency of the initial highorder model is generally consistent with the results obtained by ANSYS software. The engineering calculation efficiency is improved while the accuracy is ensured. To further simplify the model and reconstruct the onedimensional highorder model, the obtained natural frequencies are also well correlated with the shell element. The relative error of the first fifteen modes is controlled within 2.84 %, which greatly improves the accuracy compared to the traditional Timoshenko beam theory. This is due to the addition of four inplane and five outofplane highorder feature deformations on top of the low order feature deformations. They weakens the stiffness of the twincell box girde, leading to a decrease in vibration frequency. The effectiveness and accuracy of higherorder elements have been verified through comparison. When engineering calculations are considered for second or higher order natural frequencies, it is evident that the deformation of higherorder crosssectional features cannot be ignored.
Table 1The first fifteen natural frequencies and relative errors of the twincell box girder
Mode number  ANSYS  Timoshenko beam  Initial beam  Proposed beam  
${f}_{1}$ (Hz)  ${f}_{2}$ (Hz)  ${u}_{1}$ / %  ${f}_{3}$ (Hz)  ${u}_{2}$ / %  ${f}_{4}$ (Hz)  ${u}_{3}$ / %  
1  8.9692  9.1964  2.53  8.7520  –2.42  9.0650  1.07 
2  12.746  13.423  5.31  12.485  –2.05  12.916  1.33 
3  16.292  22.224  36.41  16.146  –0.9  16.541  1.53 
4  21.141  25.194  19.17  21.028  –0.53  21.736  2.81 
5  23.876  30.782  28.92  23.523  –1.48  23.982  0.45 
6  24.209  38.362  58.46  24.017  –0.79  24.273  0.27 
7  26.802  42.155  57.28  26.381  –1.57  26.59  –0.79 
8  28.271  46.813  65.59  27.905  –1.29  29.073  2.84 
9  32.744  51.369  56.88  32.326  –1.28  32.713  –0.09 
10  33.383  56.214  68.39  33.055  –0.98  33.287  –0.29 
11  33.422  60.71  81.65  33.329  –0.28  33.952  1.59 
12  36.371  73.449  101.94  35.522  –2.33  35.849  –1.44 
13  37.328  75.114  101.23  36.698  –1.69  37.166  –0.43 
14  38.104  79.819  109.48  37.357  –1.96  37.542  –1.48 
15  38.374  93.639  144.02  37.755  –1.61  38.046  –0.85 
4.2. Applicability verification
In order to further investigate the dynamic behavior of twincell box girders, a comparison is made between the 1st9th vibration modes of the reconstructed onedimensional highorder model and those of the ANSYS shell. The comparison was divided into nine pairs according to the order of the vibration modes, as shown in Fig. 4.
In each pair, the upper is derived using the reconstructed highorder model in Matlab software while the lower represents the ANSYS shell result. By comparing them, it can be observed that the good agreements between the reconstructed model and the ANSYS shell model. This indicated that the onedimensional highorder model of twincell box girder, which took into account fifteen crosssectional feature deformations, was able to accurately reproduce the threedimensional dynamic behavior of the twincell box girder. At the same time, the identified inplane and outofplane deformation modes were reflected in the vibration model, which strengthened the evidence linking deformation modes and structural behaviors. These findings further demonstrated the accuracy of the reconstructed highorder model.
The slenderness ratio of thinwalled beams may affect the modeling accuracy. As shown in Fig. 5, twincell box girders with spans of 20 m, 30 m, 40 m, and 60 m are selected for analysis to investigate the dynamic characteristics of box girders with different spans. The blue lines represent the reconstructed model, and the red imply the results of the ANSYS shell.
The natural frequency of the box girder decreases continuously as the span increases. When the span of the box girder is 20 m, the overall span to width ratio of the box girder is 2.5, and the calculated relative error can also be controlled within 5.85 %. As the span of the box girder continues to increase, the relative error also decreases. It is worth mentioning that the reconstructed model achieves a high level of accuracy while utilizing only fifteen degrees of freedom. This significantly improves the computational efficiency. This suggests that the reconstructed model still exhibits a certain correlation with the shell model despite reducing the degrees of freedom of the initial model. The identified crosssectional feature deformation is applicable to twincell box girder models with a span to width ratio of 2.5 or higher. The application scope is more broader than classical beam theory.
Fig. 4Comparison of vibration modes from reconstructed highorder model and ANSYS shell model
Fig. 5Comparison of the first fifteen frequencies between reconstructed model and ANSYS shell model
To further verify the effectiveness of higherorder deformations, the inplane and outofplane high feature deformations were sequentially removed from the reconstructed higherorder model. The calculation results were shown in Fig. 6. The $x$axis represents the onedimensional higherorder model with fourteen degrees of freedom (remove inplane or outofplane one in sequence), and the $y$axis represents the1st10th modes of the model. When the fourth outofplane warping higherorder deformation was removed, the natural frequency error solved by the model increased. Different results were obtained when other outofplane warping deformation results were removed. The fourth higher order warping deformation has a significant proportion of weight, while the latter three warping deformations have a smaller proportion of weight in the 1st10th modes. It demonstrates the importance of accurately selecting significant crosssection deformations while ensuring accuracy requirements. When the crosssectional deformation in the plane was removed, the frequency error changed greatly. It indicated that the weight of the selected inplane crosssectional deformation was relatively large in the calculation of the 1st10th modes natural frequencies. This again demonstrated the accuracy and effectiveness of higherorder feature deformation.
Fig. 6Relative errors regarding natural frequencies with inplane and outofplane feature deformations removed from the higherorder model in sequence
5. Conclusions
In this paper, a onedimensional high order model of singlebox twincell box girders was proposed with deformable crosssection. A set of shape functions was defined to capture crosssectional deformation to construct the onedimensional highorder initial model. The generalized eigenvalue problem of the governing equation was solved to uncouple the generalized displacements, and the combination of basis function weights were obtained involved with principal component analysis. In order to reduce the degree of freedom of the model and improve computational efficiency, a new shape function matrix was used to update the onedimensional highorder initial model.
Numerical example analysis shows that the reconstructed onedimensional highorder model can accurately describe the structural behavior of the twincell box girder. The calculated natural frequencies were much more accurate than classical beam theory and closer to twodimensional shell elements. At the same time, the reconstructed onedimensional highorder model had universal applicability to different constraint conditions and span to width ratios. It can also accurately reproduce 3D dynamic behaviours of the twincell box girder.
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About this article
The authors gratefully acknowledge the ﬁnancial support of National Natural Science Foundation of China (Grant No. 51805144), and Changzhou Sci and Tech Program (Grant No. CJ20220081 and CZ20190018).
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Yuhang Zhu was responsible for writing–original draft preparation and visualization. Lei Zhang was responsible for methodology and writingreview and editing. Tao Zeng was responsible for formal analysis and investigation.
The authors declare that they have no conflict of interest.