With the continuous construction of 500 m concrete-filled steel-tube (CFST) arch bridges such as the Bosideng Yangtze River Bridge and the Hejiang Changjiang Highway Bridge, the deviation between the dead pressure line and the arch axis produced by extant arch axis optimization methods increases. Therefore, an arch axis optimization method for long-span CFST arch bridges with a truss section must be designed. Following the optimization of the truss arch axis, this study develops the minimum section eccentricity method that aims to optimize the arch axis of long-span CFST arch bridges. To minimize the main tube eccentricity of the truss arch, the bending moment of the main tubes is reduced by applying the main tube eccentricity method iteratively in a finite element model. Afterward, a smooth and reasonable arch axis is fitted by applying a cubic spline interpolation function in MATLAB. The entire optimization procedure is performed using the Bosideng Yangtze River Bridge as an example. Compared with that of optimal arch axis line types (e.g., parabola and catenary) and other traditional arch axes, the bending moment of main tubes optimized by the proposed method is substantially lower and more uniformly distributed along the arch axis span. The mechanical properties of the finished bridge, including its strength, stiffness, and stability, are all improved, thereby verifying the feasibility of using the proposed method to optimize the arch axis of CFST arch bridges with a truss section.
Given their use of concrete-filled steel tubular (CFST) materials and convenient construction, CFST arch bridges have been constructed on a large scale all across China. Long-span CFST arch bridges have also been built in the country given the mature calculation theory and construction technologies used by these structures. The truss section, also known as the truss arch, serves as the main arch section of long-span CFST arch bridges. Each CSFT bridge has a unique structural form and faces an increasingly complex stress state. Apart from the geometric nonlinearity of these bridges, the deviation between their arch bridge axis and pressure line increases as the latter moves toward a circular curve, parabola, and catenary with few control parameters, which are unable to meet the structural force requirements. Therefore, to ensure a reasonable stress state for the truss arch of long-span CFST arch bridges, a reasonable arch axis must be selected [
Arcs, parabolas, catenaries, and spline curves are generally selected as arch axes of arch bridges. Arcs are derived from hydrostatic pressure lines and show huge deviations from the true dead load pressure line of an arch bridge, thereby making them applicable only to small arch bridges spanning less than 20 m. Quadratic parabolas are derived from three-hinged arches under the action of vertical uniform load. Give that the load on a CFST arch bridge is lighter than that on a masonry arch bridge (which is close to the vertical uniform load), most CFST arch bridges spanning less than 100 m use parabolas as their arch axes. Catenaries, which are used by most long-span CFST arch bridges [
Optimizing the main truss of long-span CFST arch bridges ensures that the main tubes of the truss section are in an optimal stress state. To this end, this paper proposes the minimum main tube eccentricity method as a new optimization method specifically designed for arch axes. A finite element model has also been established to treat the initial stress state of the whole structural system as the initial state of geometrically nonlinear calculation [
The rest of this paper is outlined as follows. Section
Among the members of a truss arch, main tubes bear the axial pressure along the arch axis, whereas branch tubes connect and support the main tubes while bearing the shear force and a small amount of axial force in the whole truss beam element. Therefore, when optimizing the arch axis of a truss arch, only the bending moment of main tubes needs to be considered to ensure a minimum bending moment under a dead load. To minimize the main tube eccentricity of a truss arch and subsequently optimize the arch axis of a CFST arch bridge, the maximum main tube eccentricity of each truss section should be reduced.
Following the internal force and parameters of each member as illustrated in Figures
Schematic of the truss section.
Schematic of truss arch elevation.
The main tube eccentricity of section
Therefore, to minimize
To minimize the main tube eccentricity of the truss arch, the optimization of the arch axis of a long-span CFST arch bridge needs to be analyzed. Given the complex structure of a truss arch and the uncertainty of arch axis shape, one cannot directly establish a relationship between the variations in main tube eccentricity and bending moment. Therefore, the truss arch design should be optimized by iterating the maximum main tube eccentricity of each truss section to an admissible value range [
The following assumptions are established in the calculation: According to the characteristics of the discrete structure and by assuming that the length of two adjacent main tube elements The total bending stiffness The two adjacent main tube elements
Figure
Simplified calculation model of the main tube. (a) Simplified structural diagram of the main tube. (b) Simplified bending moment diagram of the main tube. (c) Adjustment value
The bending moment
When forced displacement
The bending moment at node
By substituting formula (
The deviation coefficient of main tube eccentricity (
Using these formulas, the adjustment value of main tube eccentricity for offsetting the bending moment at node
Given that the elements of a cable system require posttensioning forces, their initial configuration under dead loading must be identified in advance. To calculate the initial stress in each cable, the cable optimization model and constrained expression are built based on the cable force optimization and constrained energy minimization methods for a cable-stayed bridge [
Given that a nonlinear behavior is expected, the cable system is simulated by the “cable element” instead of the “truss element” in the Midas finite element model. The initial tension of each cable leads to the initial stiffness of the elements of the cable system and the initial stress of the whole structure system, thereby facilitating a geometric nonlinear analysis. In the finite element model analysis, the coupling behavior between the cable and floor systems leads to each cable not in the vertical direction, and the section stiffness of the cable element is modified in time according to the current tension of the cable element. The corner of nodes at both ends of the cable element represents the rotation angle of the tangent coordinate direction of the catenary element node, which is usually inconsistent with each other. The above finite element simulation reflects reality as close as possible.
The iterative process of main tube eccentricity is illustrated in Figure
Iterative process of main tube eccentricity.
The simulated annealing algorithm is chosen to avoid trapping in local optima, which can not only iterate in the direction of objective function optimization but also accept the deterioration of objective function with a certain probability to ensure the reliability of obtaining global optimal solution [
The objective function which we set in MATLAB is as follows:
Then, a MATLAB program is presented to search for the minimum of the objective function using simulated annealing algorithm.
The iterative calculation of main tube eccentricity produces an irregular arch axis instead of a smooth curve to guide construction. To ensure the curvature aesthetics and constructability of the arch axis, the cubic spline curve is used for interpolating and fitting this axis.
According to the properties of the cubic spline curve, the second derivative
By integrating formula (
The polynomial coefficients
This research mainly focuses on the cubic spline curve interpolation fitting of the arch axis of a long-span CFST arch bridge. A half-bridge is selected as the interpolation fitting object due to the longitudinal symmetry of a CFST arch bridge. To ensure that the curvature at the midspan and arch foot of the CFST arch bridge remains unchanged, the first boundary condition of the cubic spline interpolation function is selected for the design, that is,
According to formulas ( Following the second-order continuity of the cubic spline curve at the nodes,
Therefore, the expression
Therefore, by using The formula is derived by using the first boundary condition.
By using formula ( The equation group of
Let
Using formula (
The optimization process of the cubic spline interpolation function is shown in Figure
Optimization of the cubic spline interpolation function.
The iterative main tube eccentricity method, which considers the optimization of the cubic spline interpolation function, is used to optimize the truss arch of the 530 m Bodiseng Yangtze River Bridge as an example. The structural stress states before and after the optimization of the truss arch are also compared.
As shown in Figure
Bosideng Yangtze River Bridge and its elements.
Given that the arch axis of the Bosideng Yangtze River Bridge is a catenary with an arch axis coefficient of
Iterative calculation of main tube eccentricity.
Number of iterations |
|
|
Max-My (kN·m) |
---|---|---|---|
0 | 15.00 | 63.23 | 521.53 |
1 | 15.00 | 49.76 | 365.30 |
2 | 15.00 | 41.88 | 353.89 |
3 | 15.00 | 37.14 | 359.23 |
4 | 15.00 | 31.51 | 303.95 |
5 | 15.00 | 30.39 | 261.61 |
6 | 15.00 | 25.33 | 205.62 |
7 | 15.00 | 22.11 | 141.47 |
8 | 15.00 | 19.99 | 126.38 |
9 | 15.00 | 19.74 | 113.78 |
10 | 15.00 | 17.87 | 115.00 |
11 | 15.00 | 16.58 | 113.80 |
12 | 15.00 | 16.23 | 112.83 |
13 | 15.00 | 15.53 | 112.65 |
14 | 15.00 | 15.26 | 99.44 |
15 | 15.00 | 14.91 | 100.85 |
Maximum main tube eccentricity.
Maximum bending moment of the main tube.
As can be seen from the charts above, the maximum main tube eccentricity of the Bosideng Yangtze River Bridge decreases from 63.23 mm to 14.91 mm after 15 external cycle iterations of the finite element model. Moreover, the maximum bending moment of the main tube decreases by 517.13% from 521.53 kN·m to 100.85 kN·m. The bending moment of each key section of the Bosideng Yangtze River Bridge is controlled within 100 kN·m, thereby highlighting the remarkable effect of iterative calculation.
The cubic spline interpolation function is calculated by taking the final model of the iteration calculation of main tube eccentricity as the initial structure. Six key sections, namely, the arch foot section, border of the arch girder section, L/8, L/4, 3L/8, and midspan section, are selected as the control sections of the cubic spline interpolation function. The centroid coordinates of these sections are extracted from the finite element model and presented in Table
Centroid coordinates of the six key sections.
Centroid coordinates | Arch foot | Border of arch girder | L/8 span | L/4 span | 3L/8 span | Midspan |
---|---|---|---|---|---|---|
|
−251.22 | −228.05 | −178.75 | −128.70 | −64.35 | 0.00 |
|
−105.34 | −85.93 | −51.27 | −25.57 | −5.07 | 1.98 |
According to the structural form of the Bosideng Yangtze River Bridge, the first boundary condition is selected as an additional condition of the cubic spline curve. The angles of the arch foot and midspan sections are
The coefficient matrix
The cubic spline curve of the half-bridge of the Bosideng Yangtze River Bridge is obtained based on the coefficient matrix
Half-span arch axis fitted by the cubic spline curve.
After calculating the coefficient matrix
By using main tubes 1 (upper chord) and 2 (lower chord) as examples, the internal force state of each key section of the Bodiseng Yangtze River Bridge is analyzed by comparing three types of optimal arch axis lines specific to this bridge, namely, the arch axis line type optimized by a parabola with an exponent of 2.1, the catenary with an arch axis coefficient of 1.35, and the cubic spline curve of coefficient matrix
Bending moment of the six key sections.
As indicated by the maximum bending moment of the upper and lower chords of each key section, the parabola, catenary, and cubic spline curve have maximum bending moments of 546.65, 455.39, and 233.40 kN·m, respectively.
The curve of the cubic spline interpolation function is better than those of the parabola and catenary. The bending moment values of the six key sections of the cubic spline curve at the arch foot, border of the arch girder, L/4, 3L/8, and midspan significantly decrease. The L/8 span has slightly large bending moment values yet the bending moment at the section location is small, thereby highlighting the remarkable optimization effect of the internal force state of the main arch rib.
Considering the self-weight and moving load of the structure, the maximum absolute stress of the upper and lower chords of the three optimal arch axis line types is calculated and compared under the basic load combination of the ultimate limit state according to formula (
Stress distribution of the main arch rib of the half-bridge. (a) Stress distribution of the upper chord. (b) Stress distribution of the lower chord.
Figure
The maximum vertical deflection (i.e., sum of the absolute values of maximum positive and negative deflection) under the lane load (excluding the impact coefficient) and the deflection curves of the upper and lower chords of the main arch rib are presented in Figure
Deflection distribution of the main arch rib. (a) Deflection distribution of the upper chord. (b) Deflection distribution of the lower chord.
Figure
In analyzing the elastic stability of the CFST arch rib, the load of the bridge deck system, suspender weight, and moving load are all equivalent to the concentrated force of the top node of the suspender. The equivalent concentrated force of the moving load varies, whereas that of the other loads is invariant. A buckling analysis is conducted by using the Midas/Civil spatial finite element model. The results for the first four instability modes are shown in Figure
First four instability modes. (a) First-order instability mode. (b) Second-order instability mode. (c) Third-order instability mode. (d) Fourth-order instability mode.
As shown in the figure, the first four instabilities of the Bosideng Yangtze River Bridge are transverse. The elastic stability coefficient is shown in Table
Stability coefficient of the first four instability modes.
Modal order | Instability mode | Parabola | Catenary | Spline curve |
---|---|---|---|---|
First mode | Symmetrical transverse | 18.78 | 19.01 | 19.25 |
Second mode | Antisymmetrical transverse | 22.15 | 22.73 | 22.84 |
Third mode | Symmetrical transverse | 30.17 | 31.28 | 31.28 |
Fourth mode | Antisymmetrical transverse | 35.55 | 35.74 | 35.92 |
The stability coefficients of the three optimal arch axis line types progressively decrease from parabola and catenary to cubic spline curve at any order, thereby proving that the cubic spline curve has a better stability than the other two curves.
The proposed minimum main tube eccentricity method considers the actual stress state of the main tube, effectively reduces its bending moment, and controls the maximum eccentricity to optimize the internal force state of the truss arch. This method is designed based on an iterative approach, which may not be the global optimum. Nevertheless, this approach increases the convergence of the algorithm and can be used to build an optimal solution that meets the engineering accuracy requirements. The iteration calculation is also convenient and efficient, thereby facilitating the optimization.
The proposed arch axis optimization method is applied by using the Bosideng Yangtze River Bridge as an example. The internal forces of three optimal arch axis line types are compared, and the findings indicate that the proposed method controls the eccentricity of the key sections, considerably reduces the bending moment of the main tubes, and ensures a uniform distribution of internal forces. Meanwhile, the analysis of the mechanical properties of the finished bridge state reveals that the stress amplitude of the cubic spline curve is smaller and more uniform than that of others, whereas its stiffness and stability are increased.
The testing and analysis data used to support the findings of this study are included in the article.
The authors declare no conflicts of interest regarding the publication of this paper.
This research was sponsored by the Construction Control Group of the Bosideng Yangtze River Bridge.