Calculation method on shape finding of self-anchored suspension bridge with spatial cables

Abstract

Based on the spatial model, a reliable and accurate calculation method on the shape finding of self-anchored suspension bridge with spatial cables was studied in this paper. On the principle that the shape of the main cables between hangers is catenary, the iteration method of calculating the shapes of the spatial main cables under the load of hanger forces was deduced. The reasonable position of the saddle was determined according to the shape and the theoretical joint point of the main cables. The shapes of the main cables at completed cable stage were calculated based on the unchanging principle of the zero-stress lengths of the main cables. By using a numerical method combining with the finite element method, one self-anchored suspension bridge with spatial cables was analyzed. The zero-stress length of the main cables, the position of the saddle, and the pre-offsetting of the saddle of the self-anchored suspension bridge were given. The reasonable shapes of the main cables at bridge completion stage and completed cable stage were presented. The results show that the shape-finding calculation method is effective and reliable.

Keywords

bridge engineering / self-anchored suspension bridges / special cables / shape-finding / calculation method

Introduction

The decreasing lateral stiffness of suspension bridges with the increasing of the span and the decreasing of the width-span ratio affects the dynamic stability of the bridges. The three-dimensional cable systems of suspension bridges formed from the main cables and hangers provide the increased lateral and torsional stiffness of the whole bridge [1].

The self-anchored suspension bridge does not require the massive end anchorages used for the typical earth-anchored suspension bridge. Instead, the main cables are anchored to each end of the stiffening girder. These axial forces on the stiffening truss system and the main towers result in the initial deformation of a self-anchored suspension bridge. These initial displacements of anchorages and saddles denote, from the aspect of structural analysis, changes in the boundary conditions. Therefore, even with the best analysis of initial coordinates and unstrained lengths of cables, the initial equilibrium state under full dead loads cannot be established when a self-anchored suspension bridge is being considered [2].

Two computational models were used to analyze the internal forces and shape of the self-anchored suspension bridge with spatial cables, the Yongjong Grand Bridge [3]. The whole computational procedure was very complicated, in which the effect of saddle and the harmonious condition of internal forces among the neighboring catenary sections were not taken into account, and the computational method of the shape of completed cable stage was not proposed. In view of this deficiency, with a spatial analysis model and extending the calculation method of segmental catenary from plane to spatial main cable shape, this paper presents the numerical calculation method, which is used to analyze the shape of the main cables at bridge completion stage and completed cable stage. In addition, the hanger forces not only decide the shape of the main cables but also influence the internal force distribution of the stiffening girder for the different internal forces of the self-anchored suspension bridge from that of earth-anchored suspension bridge. A numerical method combining with the finite element method is presented for solving the problem in the paper. The hanger forces are adjusted continually, and the shape of completion stage is modified repeatedly until the shape satisfies the design requirement. Then, according to the shape of completion stage, the location of the saddle on the tower is calculated, and the zero-stress length of the main cables is corrected. On the principle of the zero-stress length of the main cables, the shape of the completed cable stage is determined. Finally, one self-anchored suspension bridge is analyzed using the presented method, and the results demonstrate the correctness of the method in this paper.

Calculation method of shape of spatial main cables at bridge completion stage

Assumptions for calculation

There are three assumptions in the calculation of the main cables as follows:

1) The main cables are ideal flexible cable, which can only bear the tension, and the influence of the sectional bending stiffness is ignored.

2) The material of the main cables is linear elastic and conforms to the Hook’s law.

3) The changes of the cross section area of the main cables before and after deformation are not considered.

Numerical calculation method of spatial main cables shape

As for the single-span cable (as shown in Fig. 1), because the shape of the main cables among the neighboring hangers is catenary, the coordinate equation for certain catenary section can be formulated as follows:

(1)X={\displaystyle \frac{F_{X_{\mathrm{1}}}{S}_{\mathrm{0}}}{EA}}+{\displaystyle \frac{F_{X_{\mathrm{1}}}}{w}}\left(a\sinh {\displaystyle \frac{F_{Z_{\mathrm{1}}}}{H}}-a\sinh {\displaystyle \frac{F_{Z_{\mathrm{1}}}-w{S}_{\mathrm{0}}}{H}}\right),

(2)Y={\displaystyle \frac{F_{Y_{\mathrm{1}}}{S}_{\mathrm{0}}}{EA}}+{\displaystyle \frac{F_{Y_{\mathrm{1}}}}{w}}\left(a\sinh {\displaystyle \frac{F_{Z_{\mathrm{1}}}}{H}}-a\sinh {\displaystyle \frac{F_{Z_{\mathrm{1}}}-w{S}_{\mathrm{0}}}{H}}\right),

(3)Z={\displaystyle \frac{w{S}_{\mathrm{0}}^{\mathrm{2}}-\mathrm{2}{F}_{Z_{\mathrm{1}}}{S}_{\mathrm{0}}}{\mathrm{2}EA}}-{\displaystyle \frac{H}{w}}\left(\sqrt{\mathrm{1}+{\left({\displaystyle \frac{F_{Z_{\mathrm{1}}}}{H}}\right)}^{\mathrm{2}}}-\sqrt{\mathrm{1}+{\left({\displaystyle \frac{F_{Z_{\mathrm{1}}}-w{S}_{\mathrm{0}}}{H}}\right)}^{\mathrm{2}}}\right),

H=\sqrt{F_{X_{\mathrm{1}}}^{\mathrm{2}}+F_{Y_{\mathrm{1}}}^{\mathrm{2}}},

where F_{X_{\mathrm{1}}},F_{Y_{\mathrm{1}}},F_{Z_{\mathrm{1}}} are the three components of the internal force of the main cable, respectively; S₀ is the zero-stress length; w is the weight per unit length; and EA is the initial tensile stiffness of the cross section.

The single-span cable (as shown in Fig. 1) is divided into n+1 segments by the n hanger forces, and each segment should satisfy the catenary conditions. So the coordinate equation for i catenary section can be formulated as follows:

(4)X_i=X_{i-\mathrm{1}}+{\displaystyle \frac{F_{X_{i-\mathrm{1}}}{S}_{\mathrm{0}i}}{EA}}+{\displaystyle \frac{F_{X_{i-\mathrm{1}}}}{w}}\left(a\sinh {\displaystyle \frac{F_{Z_{i-\mathrm{1}}}}{H_{i-\mathrm{1}}}}-a\sinh {\displaystyle \frac{F_{Z_{i-\mathrm{1}}}-w{S}_{\mathrm{0}i}}{H_{i-\mathrm{1}}}}\right),

(5)Y_i=Y_{i-\mathrm{1}}+{\displaystyle \frac{F_{Y_{i-\mathrm{1}}}{S}_{\mathrm{0}i}}{EA}}+{\displaystyle \frac{F_{Y_{i-\mathrm{1}}}}{w}}\left(a\sinh {\displaystyle \frac{F_{Z_{i-\mathrm{1}}}}{H_{i-\mathrm{1}}}}-a\sinh {\displaystyle \frac{F_{Z_{i-\mathrm{1}}}-w{S}_{\mathrm{0}i}}{H_{i-\mathrm{1}}}}\right),

(6)Z_i=Z_{i-\mathrm{1}}+{\displaystyle \frac{w{S}_{\mathrm{0}i}^{\mathrm{2}}-\mathrm{2}{F}_{Z_{i-\mathrm{1}}}{S}_{\mathrm{0}i}}{\mathrm{2}EA}}-{\displaystyle \frac{H_{i-\mathrm{1}}}{w}}\left(\sqrt{\mathrm{1}+{\left({\displaystyle \frac{F_{Z_{i-\mathrm{1}}}}{H_{i-\mathrm{1}}}}\right)}^{\mathrm{2}}}-\sqrt{\mathrm{1}+{\left({\displaystyle \frac{F_{Z_{i-\mathrm{1}}}-w{S}_{\mathrm{0}i}}{H_{i-\mathrm{1}}}}\right)}^{\mathrm{2}}}\right),

H_{i-\mathrm{1}}=\sqrt{F_{X_{i-\mathrm{1}}}^{\mathrm{2}}+F_{Y_{i-\mathrm{1}}}^{\mathrm{2}}},

where F_{X_{i-\mathrm{1}}},F_{Y_{i-\mathrm{1}}},F_{Z_{i-\mathrm{1}}} are the three components of the internal force of the i catenary section, respectively.

According to the third former assumption, three components of the main cable at bridge completion stage can be expressed as

(7)F_{X_i}=F_{X_{i-\mathrm{1}}},

(8)F_{Y_i}=F_{Y_{i-\mathrm{1}}}-P_{\mathrm{3}i}(Y_i-Y_{gi})/Z_i-Z_{gi},

(9)F_{Z_i}=F_{Z_{i-1}}-P_{3i}-w{S}_{0i},

where P_3i is the vertical component of the hanger force at the i point loading on the main cable, as shown in Fig. 2; Y_gi,Z_gi are the lateral and vertical coordinates of the anchor end of the hanger at the i point, respectively.

Using the Newton’s method, the coordinates of the main cable can be obtained from Eqs. (4)-(9), which are composed of the balance coordinate equations of the elastic main cable under the loads of deadweight and hanger forces.

The shape and cable forces of the main cables for the main span are calculated first. Then, the shape and cable forces of the side spans can be obtained according to the unchanged principle of the horizontal force along the longitudinal direction of the bridge.

Shape calculation method of self-anchored suspension bridge under completed bridge stage

Although the shape of the spatial main cables under completed bridge stage by the above numerical analysis method can be calculated accurately, the huge axial pressure induced by the main cables that is anchored on the stiffening girder of the self-anchored suspension bridge will change the shape of the main cables [4,5].

Because of the special stress state of the self-anchored suspension bridge, a numerical analysis method combining with the finite element method is presented to determine the shape of the self-anchored suspension bridge under completed bridge stage in this paper. In other words, first, the hanger force is supposed to be the sum of the beam segment deadweight and the secondary dead load, and then, the initial shape of the self-anchored suspension bridge at completed bridge stage is calculated using the presented numerical analysis method. Second, the finite element model is established according to the initial shape and is analyzed under the structural deadweight. Checking the vertical deflection of the girder, if the value is in the error range, then the shape is the final bridge curve, or else, we should modify the hanger forces and calculate the shape and internal forces of the main cables again until the curve of the main girder meets the design need.

Calculation method of saddle on top of tower

For adapting to the spatial arrangement of the main cables, the shapes of the cable saddles on the top of the tower are three-dimensional in theory to ensure that the cables are tangent to the saddle in all directions. As a result of the inconvenience of the manufacturing of the spatial saddles, the ceaseless changing of the main cables during the erection stage and the unavoidable errors of erection of the main cables, the model of the saddles should be simplified properly. The saddles are considered to be in a spatial plane (i.e., the saddle plane), as shown in Fig. 3. So it only ensures that the main cables are tangent to the saddle in the vertical direction and does not ensure that they are tangent to the saddle in the horizontal direction [6].

In the design stage of the suspension bridge, the theoretical intersection point (the I_p point) of the main cables is usually confirmed first. Then, the reasonable position of the saddle is confirmed according to the shape of the main cables. The concrete calculation method is listed as follows.

1) The coordinates (x₀, y₀, z₀) of the I_p point and the left calculation radius and the right calculation radius of the saddles (R₁ and R₂) are known.

2) The two tangent points on the main cables, T₁(x₁,y₁,z₁) and T₂(x₂,y₂,z₂), are assumed or adjusted, and then, the two points and the point, I_p, constitute the saddle plane. Accordingly, the normal vector of the saddle plane, {m₀,n₀,z₀},could be obtained and expressed as

(10)\{\begin{array}{l}{m}_0=(y_1-y_0)(z_2-z_0)-(y_2-y_0)(z_1-z_0),\hfill \\ n_0=(x_2-x_0)(z_1-z_0)-(x_1-x_0)(z_2-z_0),\hfill \\ l_0=(x_1-x_0)(y_2-y_0)-(x_2-x_0)(y_1-y_0).\hfill \end{array}

3) The two angles,α₁ and α₂, between the facade projections of the main cables and the horizontal line at the two points of T₁ and T₂ are calculated.

4) If the plane (defined as the normal plane 1) goes through the point T₁ and is perpendicular to the saddle plane and the angle between the facade projection of the normal vector of the plane and the horizontal line is α₁, the normal vector of the plane is {1,tanα₁,tanβ₁}, as shown in Fig. 3(b), where tanβ₁=-(m₀+n₀tanα₁) /l₀. Similarly, if the plane (defined as the normal plane 2) goes through the point T₂ and is perpendicular to the saddle plane and the angle between the facade projections of the normal vector of the plane and the horizontal line is α₂, the normal vector of the plane is {1,tanα₂,tanβ₂}, where tanβ₂=-(m₀+n₀tanα₂)/l₀.

5) According to the normal vectors of the normal plane 1 and the saddle plane and to the point T₁, which belongs to not only the normal plane 1 but also the saddle plane, the intersection line equation of the two planes could be obtained. Similarly, the intersection line equation of the normal plane 2 and the saddle plane could be obtained. Thus the coordinates of the intersection point of the two spatial lines could be easily obtained, that is, the coordinates of the center of the circle.

6) Calculate the distances between the center of the circle and the two tangent points and compare the distances with the given saddle radius. If the error is smaller than the permitted error, the result is convergent, or else, return to the step 2) and continue calculating until the result is convergent.

Calculation of shape finding at completed cable stage

According to the unchanging principle of the zero-stress length of the main cables, the shape of the main cable at completed cable stage is calculated, and the pre-offsetting of the saddle is figured out to make the main cables at both sides of the saddle meet the force balance requirements. So the calculation methods for each span are the same. When the effect of the saddle is not considered, the force components of one end of the cable can be assumed first, and then, the shape of the main cable is calculated from left to right and segment by segment until the calculated coordinates of the other end of the main cable meet the convergent requirement. When the effect of the saddle is taken into account, the tangent points between the main cables and the saddle can be assumed first, and then, the cable forces at the tangent points are calculated. Thus, the zero-stress length inside the saddle could be obtained. The shape of the main cable between the two tangent points is calculated according to the method without considering the influence of the saddle [7,8].

Saddle is a device, which veers the main cable. Although the size of the saddle is small compared with the bridge span, the saddle constrains the deflections of the main cable directly and makes the cable to be tangent to it. The theoretical intersection point (the I_p point) is a virtual point and is commonly referred to as the intersection point of the main cable under the completed bridge stage. If the shape of the cable at completed cable stage is still calculated according to the I_p point, the accurate relative relationship between the main cable and the saddle cannot be guaranteed, and the main cable will intersect with the saddle or disengage from it. Consequently, the shape of the main cable at completed cable stage is calculated and must consider the influence of the saddle. Consequently, the influence of the saddle must be taken into account for the shape calculation at completed cable stage. Figure 4 shows the flow diagram of shape calculation at completed cable stage.

Illustrative examples

Engineering background of a self-anchored suspension bridge

The span arrangement of a self-anchored suspension bridge with spatial main cables is 35 m+77 m+60 m+248 m+35 m, and the rise-span ratio of the main span is 1/12.34, as shown in Fig. 5. The girder of the bridge is divided into two branches interconnected by cross tie beams. The steel box girder is used for the main span, and the concrete box girder is used for the side span. The main tower with the type of single column is located in the middle position of the two girders across the bridge. The main cables are divided into two portions in the direction across the bridge. The main cables of the side span are anchored to the middle position of the crossbeam and are arranged parallel, whereas the main cables of the main span are anchored to the two ends of the crossbeam and are arranged spatially.

Finite element model of self-anchored suspension bridge

Based on the program NACS, which is used for the nonlinear static and dynamic analysis of the long-span bridges, the finite element model of the self-anchored suspension bridge with 383 elements and 279 nodes is set up. The main cables and hangers are simulated by the spatial cable elements, and the tower and girder are simulated by the spatial beam elements. The bottom of the tower is fixed, the general supporting conditions are used on the piers, and the main cables are fixed on the top of the tower. The coupled connections between the tower and the girder in the vertical direction of the bridge and the direction across the bridge are used, and the elastic connection is used to simulate the damper in the direction along the bridge.

Results of main cable shape and saddle pre-offsetting

Based on the numerical method combining with the finite element method, the reasonable hanger forces are calculated, and the main cable shapes and the zero-stress length are obtained accordingly. The coordinates of the circle center of the saddle and the tangent points between the main cables and saddle are calculated according to the main cable shapes at completed bridge stage. The zero-stress lengths of the main cables are modified with consideration of the effect of the saddle. The main cable shapes, the tangent points between the main cables and saddle, and the saddle pre-offsetting at completed cable stage are calculated according to the unchanging principle of the zero-stress length of the main cables. The zero-stress lengths of the main cables are listed in Table 1. Table 2 shows the position of the saddle at completed bridge stage. Table 3 shows the tangent points between the main cables and the saddle and the pre-offsetting at the completed cable stage. The main cable shapes at completed bridge stage and the completed cable stage are listed in Table 4. The node numbering of the finite element model is show in Fig. 6.

The maximum deflection of the girder at final completed bridge stage is 2 cm and satisfies the design requirement of the girder shape. In addition, the pre-offsetting of the saddle is calculated by the taking-away method [9] of the finite element method, and the value of -0.129 cm agrees well with the result obtained by the presented method in this paper. It validates the validity and the feasibility of the presented method in the paper.

Conclusions

First, on the principle that the shape of the main cables between hangers is catenary, the iteration method of calculating the coordinates of the spatial main cables under the action of hangers is deduced. Then, based on the presented numerical method combining with the finite element method, the reasonable hanger forces under the completed bridge stage are determined. The shapes of the main cables at the completed bridge stage are calculated according to the hanger forces. Finally, through the calculation analysis of the bridge, the maximum deflection of the main girder is 2 cm and satisfies the design requirement. The pre-offsetting of the saddle is calculated by the taking-away method of the finite element method, and the value of -0.129 cm agrees well with the result obtained by the presented method in this paper. Through the example analysis, it shows that the presented method for calculating the main cable shapes of the self-anchored suspension bridge with the spatial main cables is validated and feasible. The method in this paper has very high calculation precision and is very easy to use. Meanwhile, the method can consider the saddle effect and make the calculation model to be closer to the actual structure. The presented method in this paper can be referenced for the fields of design, construction, and scientific research.

References

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