An analytical algorithm for the pylon saddle pushing stage and distance during the suspension bridge construction

Abstract

During the construction of suspension bridges, the stress state of the pylon (tower) is improved by pushing the pylon saddle by an appropriate distance at the proper time. An analytical algorithm for the assessment of the required timing and displacements for the pylon saddle pushing at particular construction stages is proposed and verified in this study. The timing calculation is based on the assessment of current hanger tensile forces at each construction stage and the pylon stress state, while the pushing distance/displacement is derived from the conditions of elevation difference closure and the conservation of unstrained length of the main cable segments. This algorithm was successfully applied during the construction of a particular suspension bridge in China with a main span of 730 m. The results obtained strongly indicate that the bending moment in the pylon bottom is contributed by both horizontal and vertical forces of the main cable. The horizontal constituent is dominant and its share gradually increases in the bridge construction process. In a suspension bridge with side spans of various lengths, the stresses in the pylon bottom on the side with a larger side span is more likely to exceed the limit. Therefore, the respective strength criterion controls the pylon saddle-pushing schedule. The proposed analytical algorithm is quite straightforward and is recommended for wider application.

Keywords

main cable pushing distance pushing timing pylon saddle stress suspension bridge

Introduction

Pylon (or tower) saddles are vital components of suspension bridges that provide the load transfer between the main cable and pylon. During the hoisting of a stiffening girder, it is expedient for the pylon saddle to have a free sliding bottom, while the pylon should bear the total eccentric vertical load and exhibit no displacement. However, during the actual construction process, in the absence of any constraints between the pylon saddle and pylon top, the pylon saddle would be prone to uncontrollable large-scale sliding. To avoid this, the pylon saddle is temporarily connected to the pylon top during the construction. However, during the hoisting of the stiffening girder, as the number of girder segments in the central span increases, the horizontal component of force acting on the main cable in the central span will gradually rise. As a result, the pylon top will be subjected to unbalanced forces and get inclined toward the central span. If the pylon top displacement exceeds a certain level, the pylon top may fracture. Therefore, the structure safety requires the pylon deviation rectification during the hoisting of stiffening girders. The available deviation rectification methods can be reduced to following three main ones.

Method 1. Before erecting the main cable, the pylon saddle is installed with a preliminary offset toward the side span relative to the pylon center (Gil and Choi, 2001; Hasegawa et al., 1995; Takena et al., 1992; Wang et al., 2016). During the hoisting of stiffening girders and construction of the bridge deck, the pylon saddle is gradually pushed toward the central span (Karoumi, 1999). The tensile force of the main cable is adjusted by changing the span length so that the horizontal forces acting on the both sides of the pylon will recover their equilibrium. This method will also change the pylon bending loading to the compressive vertical one. As the stiffening girders are hoisted to reach the completed bridge state, the vertical loading conditions of the pylon, as required in the completed bridge state, will be finally obtained.

Method 2. Before erecting the main cable, several steel wire ropes are connected to the pylon top, and they are pulled toward the side spans using tensioning equipment. Therefore, the pylon will be inclined toward the side spans by an appropriate distance (Jensen and Petersen, 1994). During the hoisting of stiffening girders, the tension in the steel wire ropes is gradually released. Consequently, the inclination distance of the pylon decreases until the required vertical state in the completed bridge state is reached.

Method 3. Before erecting the main cable, the saddle is installed at the center of the pylon and bound to the pylon. During the hoisting of stiffening girders, the cable strands of the side spans are tensioned in stages with tensioning equipment. The tension of the main cable in the side spans is adjusted manually so that the horizontal tensions of the main cable in the central span and side spans are in balance. During the tensioning process, the pylon gradually returns to the vertical state from the state of bending toward the central span. In this way, the deviation rectification is performed for the pylon.

The analysis of the above three methods of deviation rectification shows that the first one is the most easy to implement and control. Therefore, it is preferable by most designers and finds wide applications. Its further refinement is proposed in this article. Noteworthy is that during the hoisting of stiffening girder in a suspension bridge, the main cable is subjected to the most significant and frequent load variations. For this reason, the pylon undergoes a constant longitudinal displacement, and the major part of pylon saddle pushing has to be finished at this construction stage. To ensure an adequate stress state of the pylon in the construction process, the calculation of the timing and distance of the pylon saddle pushing is the most critical monitoring target during the hoisting of stiffening girder.

The finite element method (FEM) is commonly used for calculating the pylon saddle pushing of the suspension bridge. He et al. (2007) elaborated the 3D FEM model for the entire Yichang Yangtze Highway Bridge in China using the degenerated solid elements based on the traditional 3D solid isoparametric elements. Then, the stress evolution in the pylon during the hoisting of the stiffening girder was simulated. Finally, the principle of saddle pushing by frequent short steps was proposed, which implied that the number of pushing steps was increased, while the distance acquired by each pushing was reduced. This was conducive to enhancing the pylon safety margin and the entire bridge safety. Zhang and Wang (2017) simulated the process of pylon saddle pushing by using the cooling method. Briefly, the temperature-dependent rod elements were incorporated into the FEM model to simulate a short-step cooling of the pre-offset of pylon saddle to zero, in order to realize the pylon saddle pushing. Li et al. (2011) calculated the moment and distance of pylon saddle pushing by FEM. Moreover, the pylon stresses and longitudinal deviations were measured during the bridge construction. The adjustment scheme that prioritized the pylon stress control and treated its deviation control as a secondary task was developed. The pushing distance was corrected for the pylon saddle according to the specific construction conditions at each stage.

However, some deficiencies of FEM were revealed during its application to pylon saddle pushing calculations for suspension bridges. Thus, FEM provides the approximate numerical simulation of the bridge construction process based on the particular construction features, loads, and boundary conditions. However, the actual construction process is full of uncertainties due to non-linear factors, such as large displacement of the suspension bridge and high initial internal force. As a result, the account of the above factors in numerical simulation model may become very problematic and time-consuming and even lead to large errors. To assess the cable shape in the completed bridge state and intermediate construction stages, pilot calculations and iterations are usually required through repeated forward and backward analyses, which can be quite cumbersome and complicated (Chen et al., 2015).

In this respect, analytical methods that are based on clearly defined physical principles have the advantages of high computational accuracy and speed, excellent generality, and applicability. The analytical method viability for the calculation of the main cable shape in suspension bridges was proved in several studies (Jung et al., 2015; O’Brien and Francis, 1964). However, to the best of the authors’ knowledge, no analytical algorithm for the pylon saddle pushing assessment in the suspension bridge has been reported yet.

In the present study, an analytical algorithm for determining the timing and distance of the pylon saddle pushing in the suspension bridge is proposed. This method is then applied to the pylon saddle pushing of a suspension bridge with a main span of 730 m. The calculation results verify the feasibility and effectiveness of the proposed method.

General principles

The flowchart of the proposed algorithm is shown in Figure 1. First, the shape of the main cable in the completed bridge state is calculated based on the hanger tensile force assessment combined with the elevation difference closure condition. Then, the unstrained length of the main cable is estimated, which is used as the prerequisite for the subsequent computation. After that, the free cable shape is analyzed based on the conditions of elevation difference closure and conservation of the unstrained length of main cable segments. The initial pre-offset is obtained for the pylon saddle. Next, upon the stress state analysis of the pylon, the hanger tensile forces at each construction stage are utilized to calculate the exact timing of pylon saddle pushing, that is, when the pylon saddle should be pushed at each particular construction stage. Finally, the distance of the pylon saddle pushing is calculated based on the conditions of elevation difference closure and conservation of the unstrained length of main cable segments.

Figure 1.

Flowchart of the general calculation procedure.

Hanger tensile forces in the completed bridge state

As shown in Figure 2, the lengths of the left-side, central, and right-side spans are L_l, L, and L_r, respectively. Hanger tensile forces in the completed bridge stage can be derived using the continuous multiple-rigid-support beam (Figure 3). That is, by replacing each hanger with a rigid support under the stiffening girder, the support reaction of each rigid support can be treated as a hanger tensile force under the dead weight (Jung et al., 2013; Kim and Lee, 2001; Thai and Choi, 2013).

Figure 2.

Schematic for the suspension bridge.

Figure 3.

Cable shape and hanger tensile forces of the central span.

Unstrained length of the main cable

Central span

The shape assessment of the main cable in the completed bridge state usually starts from the central span. Due to the hanger tensile forces, the shape of the main cable in the central span is a cornered multi-segment catenary, which implies that the cable shape between two adjacent hanging points is a catenary (Figure 3). Multiple coordinate systems are constructed, with point B and each particular hanging point as their origin, respectively. In these systems, each X-axis is extended horizontally to the right, while each Y-axis is extended vertically downward. Then, the catenary equation for any main cable segment takes the following form

y = c [\cosh (\frac{x}{c} + a_{i}) - \cosh a_{i}]

(1)

where c = –H/q, H is the horizontal force of the main cable (kN); q is the dead weight intensity of the main cable in the completed bridge state (kN/m); a_i is the parameter in the catenary equation for the i-th segment.

Two conditions of the elevation difference closure: (1) between the pylon-top and the mid-span point, and (2) between two pylon-tops are used to derive the respective two equations

\sum_{i = 1}^{m} Δ h_{i} = h_{B} - h_{M}

(2-1)

\sum_{i = 1}^{n + 1} Δ h_{i} = h_{B} - h_{C}

(2-2)

where m is the number of catenary segments between point B and midspan M; n is the number of hangers;n + 1 is the number of catenary segments of the main cable; Δh_i is elevation difference between two ends of any catenary segment; h_B, h_M, and h_C are the elevations of points B, M, and C, respectively, in the geodetic coordinate system (which values are known).

The above equations feature two unknown quantities, namely, the horizontal force H in the main cable and the parameter a₁ in the first catenary equation. All other terms in these equations can be expressed as the functions of these two unknown parameters.

The elevation difference Δh_i between both ends of any catenary segment is expressed as

Δ h_{i} = y (l_{i}) - y (0) = c [\cosh (\frac{l_{i}}{c} + a_{i}) - \cosh a_{i}]

(3)

where l_i is the horizontally projected length of any catenary segment of the main cable (Figure 3).

Based on the balance of forces in any hanging point, the following recurrence relation between a_i ₊ ₁ and a_i was introduced in (Zhang et al., 2018b)

{a_{i +}}_{1} = asinh [\sinh (\frac{l_{i}}{c} + a_{i}) - \frac{P_{i}}{H}], i = 1, 2, \dots, n

(4)

where P_i is the hanger tensile force of the i-th hanger.

By substituting formulas (3) and (4) into (2) and re-arranging the terms from the right part of the equation to its left one, the following two dependences for H and a₁ are obtained

\begin{matrix} f_{1} (H, a_{1}) = c \sum_{i = 1}^{m} [\cosh (\frac{l_{i}}{c} + a_{i}) - \cosh a_{i}] \\ - h_{B} + h_{M} = 0 \end{matrix}

(5-1)

\begin{matrix} f_{2} (H, a_{1}) = c \sum_{i = 1}^{n + 1} [\cosh (\frac{l_{i}}{c} + a_{i}) - \cosh a_{i}] \\ - h_{B} + h_{C} = 0 \end{matrix}

(5-2)

This system of equations can be solved by the nonlinear programming method using H and a₁ as variable via the following objective function

\sum_{i = 1}^{2} f_{i}^{2} (H, a_{1}) = 0

(6)

Using the generalized reduced gradient (GRG) nonlinear solver (Lasdon et al., 1974, 1978), which is now available in the Microsoft Excel, two unknown values (H and a₁) are derived. Then, the shape and internal force of the main cable in the central span are calculated.

The unstrained length S_i of any catenary segment of the main cable can be derive as (Zhang et al., 2018b)

\begin{matrix} S_{i} = c [\sinh (\frac{l_{i}}{c} + a_{i}) - \sinh a_{i}] - \frac{H}{2 EA} \\ {l_{i} + \frac{c}{2} [\sinh 2 (\frac{l_{i}}{c} + a_{i}) - \sinh 2 a_{i}]} \end{matrix}

(7)

where E and A are the elastic modulus and cross-sectional area of the main cable, respectively.

Side spans

The shape assessment of the main cable in side spans in the completed bridge state is performed after that of the central span using the same calculation method. The only difference is in slightly changed known conditions: the sag-to-span ratio and mid-span point elevation of the main cable are known for the main span, but not for the side spans. However, the horizontal force of the main cable in the side spans can be obtained based on the equilibrium conditions for the pylon top. It is generally assumed that the pylon bears no horizontal load of the main cable. Therefore, in side span calculations, it is considered that the horizontal force of the main cable is equivalent to that of the main span. The above parameter value is added to the known conditions for the shape analysis of the main cable in the side spans. For the sake of brevity, the further calculations are given only for the left-side span, those for the right-side span being similar.

Insofar as the left-side span has no hangers, the main cable forms a catenary. As in the “Central span” subsection, the coordinate system is constructed with point B as its origin, the X- and Y-axis being extended horizontally to the left and vertically downward, respectively.

Based on the closure of the elevation difference between points A and B, the following governing equation is constructed

c [\cosh (\frac{L_{l}}{c} + a_{l}) - \cosh a_{l}] = h_{B} - h_{A}

(8)

where a_l is the catenary equation parameter; h_A is the elevation of point A in the geodetic coordinate system.

The value of a_l is derived by solving equation (8), which yields the shape of the main cable in the left-side span.

The unstrained length S_l of the main cable in the left-side span is (Zhang et al., 2018b)

\begin{matrix} S_{l} = c [\sinh (\frac{L_{l}}{c} + a_{l}) - \sinh a_{l}] - \frac{H}{2 EA} \\ {L_{l} + \frac{c}{2} [\sinh 2 (\frac{L_{l}}{c} + a_{l}) - \sinh 2 a_{l}]} \end{matrix}

(9)

In case of the right-side span, the coordinate system is constructed with point C as the origin, the X- and Y-axis being extended horizontally to the right, and vertically downward, respectively. Then, the unstrained length S_r of the main cable in the right-side span can be derived via the same method.

Initial pre-offset of the saddle

To assess the free cable shape, the system of equations for three spans has to be constructed and solved. It contains six unknown parameters, namely, horizontal force H₀ of the free cable (which value is equal for all three spans), parameters a_l,0, a₀, and a_r,0 of the catenary equations for the left-side, central, and right-side spans, respectively; initial pre-offsets Δ_l,0 and Δ_r,0 for the left and right pylon saddles toward the respective side spans, as shown in Figure 4. Using the pre-offsets Δ_l,0 and Δ_r,0, one can convert the left-side, central, and right-side span lengths to L_l – Δ_l,0, L + Δ_l,0 + Δ_r,0, and L_r – Δ_r,0, respectively, as shown in Figure 4.

Figure 4.

Shape of the free cable.

As in the “Central span” subsection, the coordinate systems are constructed for the left-side, main, and right-side spans with points B, B, and C as the origin, respectively, the X-axis being extended horizontally to the left, right, and right, respectively, and the Y-axis vertically downward. Based on the conditions of elevation difference closure between the two ends of the main cable in each span, the following equations are established for the left-side, central, and right-side spans, respectively

c_{0} [\cosh (\frac{L_{l} - Δ_{l, 0}}{c_{0}} + a_{l, 0}) - \cosh a_{l, 0}] = h_{B} - h_{A}

(10-1)

c_{0} [\cosh (\frac{L + Δ_{l, 0} + Δ_{r, 0}}{c_{0}} + a_{0}) - \cosh a_{0}] = h_{B} - h_{C}

(10-2)

c_{0} [\cosh (\frac{L_{r} - Δ_{r, 0}}{c_{0}} + a_{r, 0}) - \cosh a_{r, 0}] = h_{C} - h_{D}

(10-3)

where c₀ = –H₀/q₀ and H₀ is the horizontal force of the free cable (kN); q₀ is the dead weight intensity of the main cable in the construction process (kN/m); h_D is the elevation of point D in the geodetic coordinate system.

For the condition of conservation of unstrained length of the main cable in each span, the following equations are established for the left-side, central, and right-side spans, respectively

\begin{matrix} c_{0} [\sinh (\frac{L_{l} - {Δ_{l}}_{, 0}}{c_{0}} + a_{l, 0}) - \sinh a_{l, 0}] - \frac{H_{0}}{2 EA} \\ {L_{l} - {Δ_{l}}_{, 0} + \frac{c_{0}}{2} [\sinh 2 (\frac{L_{l} - {Δ_{l}}_{, 0}}{c_{0}} + a_{l, 0}) - \sinh 2 a_{l, 0}]} = S_{l} \end{matrix}

(11-1)

\begin{matrix} c_{0} [\sinh (\frac{L + Δ_{l, 0} + Δ_{r, 0}}{c_{0}} + a_{0}) - \sinh a_{0}] - \frac{H_{0}}{2 EA} \\ {L + Δ_{l, 0} + Δ_{r, 0} + \frac{c_{0}}{2} [\sinh 2 (\frac{L + Δ_{l, 0} + Δ_{r, 0}}{c_{0}} + a_{0}) - \sinh 2 a_{0}]} \\ = \sum_{i = 1}^{n + 1} S_{i} \end{matrix}

(11-2)

\begin{matrix} c_{0} [\sinh (\frac{L_{r} - Δ_{r, 0}}{c_{0}} + a_{r, 0}) - \sinh a_{r, 0}] - \frac{H_{0}}{2 EA} \\ {L_{r} - Δ_{r, 0} + \frac{c_{0}}{2} [\sinh 2 (\frac{L_{r} - Δ_{r, 0}}{c_{0}} + a_{r, 0}) - \sinh 2 a_{r, 0}]} \\ = S_{r} \end{matrix}

(11-3)

The system of equations (10) and (11) is solved by the GRG nonlinear method. Thus, the abovementioned six unknown parameters are derived.

Pushing timing of pylon saddle

After installing the main cable and hangers, the hoisting of stiffening girders and bridge deck construction should start. At this stage, the pylon saddle is pushed at the appropriate time toward the central span, and then the pylon saddle is temporarily bound to the pylon. Hoisting of stiffening girders and bridge deck construction can be split into multiple steps, represented by the symbol j. Calculation of the pushing moment of pylon saddle requires the solution of simultaneous equations for the three spans. At the j-th step, there are n+7 unknown quantities, where H_l,j, H_j, and H_r,j are the horizontal forces of the main cable in the left-side, central, and right-side spans, respectively; a_l,j and a_r,j are the parameters of catenary equations of the main cable in the left- and right-side spans, respectively; a_1,j are the parameters of the catenary equation of the first catenary segment of the main cable in the central span; l_i,j is the horizontally projected length of the i-th catenary segment of the main cable in the central span (Figure 5), i=1, 2, …, n+1.

Figure 5.

Shape of the main cable and bending deformation of pylons during the construction process.

As in the “Unstrained length of the main cable” section, the coordinate systems are constructed for the left-side, main, and right-side spans with points B, B and each hanging point, and point C as the origin, respectively, the X-axis being extended horizontally to the left, right, and right, respectively, and the Y-axis vertically downward. Similarly, the conditions of elevation difference closure for the left-side, central, and right-side spans yield the following equations

c_{l, j} [\cosh (\frac{L_{l} - Δ_{l, j} + δ_{l, j}}{c_{l, j}} + a_{l, j}) - \cosh a_{l, j}] = h_{B} - h_{A}

(12-1)

\sum_{i = 1}^{n + 1} c_{j} [\cosh (\frac{l_{i, j}}{c_{j}} + a_{i, j}) - \cosh a_{i, j}] = h_{B} - h_{C}

(12-2)

c_{r, j} [\cosh (\frac{L_{r} - Δ_{r, j} + δ_{r, j}}{c_{r, j}} + a_{r, j}) - \cosh a_{r, j}] = h_{C} - h_{D}

(12-3)

where c_l,j = –H_l,j/q₀, c_j = –H_j/q₀, c_r,j = –H_r,j/q₀, H_l,j, H_j, and H_r,j are the horizontal forces (kN) of the main cable in the left-side, central, and right-side spans, respectively; Δ_{l, j} and Δ_{r, j} are the current pre-offsets for the left and right saddles, respectively, as shown in Figure 5, while δ_l,j and δ_r,j are the displacements of the left and right pylon tops, respectively, which are caused by the unbalanced horizontal forces of the main cable in the central and two side spans. The direction toward the central span being taken as positive, the respective displacements can be derived by the following formulas

{δ_{l}}_{, j} = (H_{j} - H_{l, j}) ξ_{l}

(13-1)

{δ_{r}}_{, j} = (H_{j} - H_{r, j}) ξ_{r}

(13-2)

where ξ_l and ξ_r are the flexibility coefficients (m/kN) of the left and right pylons, respectively. Noteworthy is that the flexibility coefficient represents the displacement of a bare-pylon top caused by a unit horizontal force.

Based on the condition of conservation of unstrained length of the main cable, the following equations are established for the left-side, central, and right-side spans, respectively

\begin{matrix} c_{l, j} [\sinh (\frac{L_{l} - Δ_{l, j} + δ_{l, j}}{c_{l, j}} + a_{l, j}) - \sinh a_{l, j}] - \frac{H_{l, j}}{2 EA} {L_{l} - Δ_{l, j} + δ_{l, j} + \frac{c_{l, j}}{2} [\sinh 2 (\frac{L_{l} - Δ_{l, j} + δ_{l, j}}{c_{l, j}} + a_{l, j}) - \sinh 2 a_{l, j}]} = S_{l} \end{matrix}

(14-1)

c_{j} [\sinh (\frac{l_{i, j}}{c_{j}} + a_{i, j}) - \sinh a_{i, j}] - \frac{H_{j}}{2 EA} {l_{i, j} + \frac{c_{j}}{2} [\sinh 2 (\frac{l_{i, j}}{c_{j}} + a_{i, j}) - \sinh 2 a_{i, j}]} = S_{i}, i = 1, 2, \dots, n + 1

(14-2)

c_{r, j} [\sinh (\frac{L_{r} - Δ_{r, j} + δ_{r, j}}{c_{r, j}} + a_{r, j}) - \sinh a_{r, j}] - \frac{H_{r, j}}{2 EA} {L_{r} - Δ_{r, j} + δ_{r, j} + \frac{c_{r, j}}{2} [\sinh 2 (\frac{L_{r} - Δ_{r, j} + δ_{r, j}}{c_{r, j}} + a_{r, j}) - \sinh 2 a_{r, j}]} = S_{r}

(14-3)

Given the force equilibrium in any hanging point, the following recurrence relation between values of a_i₊_{1, j} and a_i,j is obtained (Zhang et al., 2018b)

{a_{i +}}_{1, j} = asinh [\sinh (\frac{l_{i, j}}{c_{j}} + a_{i, j}) - \frac{P_{i, j}}{H_{j}}], i = 1, 2, \dots, n

(15)

where P_i,j is the hanger tensile force after the completion of the j-th construction step. During the construction process, temporary hinges between installed segments of the stiffening girder are usually adopted. Thus, the value of P_i,j is readily assessed.

The condition of the span length closure of the central span should be also satisfied. That is, the sum of horizontally projected lengths of each catenary segment should be equal to the span length

\sum_{i = 1}^{n + 1} l_{i, j} = L + Δ_{l, j} + Δ_{r, j} - δ_{l, j} - δ_{r, j}

(16)

The GRG nonlinear solver is applied to the system of equations (12), (14), and (16). Thus, the above n+7 unknown quantities are derived.

Stresses σ_l,j and σ_r,j at the bottom of the left and right pylons are obtained, respectively, by the following equations

\begin{matrix} σ_{l, j}^{max} \\ σ_{l, j}^{min} \end{matrix} = \frac{F_{l, j} + G_{l}}{A_{l}} \pm \frac{(H_{j} - H_{l, j}) h_{l} - F_{l, j} (Δ_{l, j} - δ_{l, j})}{W_{l}}

(17-1)

\begin{matrix} σ_{r, j}^{max} \\ σ_{r, j}^{min} \end{matrix} = \frac{F_{r, j} + G_{r}}{A_{r}} \pm \frac{(H_{j} - H_{r, j}) h_{r} - F_{r, j} (Δ_{r, j} - δ_{r, j})}{W_{r}}

(17-2)

where G, h, A, and W are the pylon’s dead weight, height, cross-sectional area of the bottom, and section modulus in bending of the bottom, respectively. Subscripts l and r indicate the left and right pylon, respectively; superscripts max and min of stress σ indicate the maximum and minimum values, respectively; F_l,j and F_r,j are the vertical forces exerted by the main cable on the left and right pylon tops, respectively, and are calculated as follows

F_{l, j} = H_{l, j} \sinh a_{l, j} + H_{j} \sinh a_{1, j}

(18-1)

F_{r, j} = - H_{j} \sinh (\frac{l_{n + 1, j}}{c_{j}} + a_{n + 1, j}) + H_{r, j} \sinh a_{r, j}

(18-2)

The criteria of no tensile stress at the pylon bottom and compressive stress not exceeding the material compressive strength limit are used for determining the pushing moment of saddle. If the stress at the j-th construction step satisfies this condition, in contrast to that at the next (j+1)-th construction step, then the saddle should be pushed after the completion of the j-th construction step

σ_{μ, ν}^{min} \geq 0

(19-1)

σ_{μ, ν}^{max} \leq σ_{limit}

(19-2)

where subscript μ represents l or r, superscript ν represents j or j+1, while σ_limit is the material compressive strength limit.

Pushing distance of pylon saddle

After the j-th construction step is completed, the appropriate pushing displacements (distances) in the central span direction should be realized in the left and right saddles, respectively, so that the pylons become erect (Figure 6). At this moment, the horizontal forces of the main cable are equal in the three spans. There are n+7 unknown quantities in total: $Δ_{l, j}^{'}$ and $Δ_{r, j}^{'}$ are the residual pre-offsets after pushing the left and right saddles; $H_{j}^{'}$ is the horizontal force of the main cable; $a_{l, j}^{'}$ and $a_{r, j}^{'}$ are the parameters of catenary equations for the main cable in the left- and right-side spans, respectively; $a_{1, j}^{'}$ is the parameter of catenary equation for the first segment of the main cable in the central span; $l_{i, j}^{'}$ is the horizontally projected length of the i-th main cable segment in the central span (Figure 6), i = 1, 2, …, n+1.

Figure 6.

Shape of the main cable after pushing the saddles.

Given the closure conditions of elevation difference, the following equations are established for the left-side, central, and right-side spans, respectively

c_{j}^{'} [\cosh (\frac{L_{l} - Δ_{l, j}^{'}}{c_{j}^{'}} + a_{l, j}^{'}) - \cosh a_{l, j}^{'}] = h_{B} - h_{A}

(20-1)

\sum_{i = 1}^{n + 1} c_{j}^{'} [\cosh (\frac{l_{i, j}^{'}}{c_{j}^{'}} + a_{i, j}^{'}) - \cosh a_{i, j}^{'}] = h_{B} - h_{C}

(20-2)

c_{j}^{'} [\cosh (\frac{L_{r} - Δ_{r, j}^{'}}{c_{j}^{'}} + a_{r, j}^{'}) - \cosh a_{r, j}^{'}] = h_{C} - h_{D}

(20-3)

where $c_{j}^{'} = - H_{j}^{'} / q_{0}$ .

Based on the condition of conservation of unstrained length of the main cable, the following equations hold for the left-side, central, and right-side spans, respectively

\begin{matrix} c_{j}^{'} [\sinh (\frac{L_{l} - Δ_{l, j}^{'}}{c_{j}^{'}} + a_{l, j}^{'}) - \sinh a_{l, j}^{'}] - \frac{H_{j}^{'}}{2 EA} \\ {L_{l} - Δ_{l, j}^{'} + \frac{c_{j}^{'}}{2} [\sinh 2 (\frac{L_{l} - Δ_{l, j}^{'}}{c_{j}^{'}} + a_{l, j}^{'}) - \sinh 2 a_{l, j}^{'}]} \\ = S_{l} \end{matrix}

(21-1)

\begin{matrix} c_{j}^{'} [\sinh (\frac{l_{i, j}^{'}}{c_{j}^{'}} + a_{i, j}^{'}) - \sinh a_{i, j}^{'}] - \frac{H_{j}^{'}}{2 EA} \\ {l_{i, j}^{'} + \frac{c_{j}^{'}}{2} [\sinh 2 (\frac{l_{i, j}^{'}}{c_{j}^{'}} + a_{i, j}^{'}) - \sinh 2 a_{i, j}^{'}]} \\ = S_{i}, i = 1, 2, \dots, n + 1 \end{matrix}

(21-2)

\begin{matrix} c_{j}^{'} [\sinh (\frac{L_{r} - Δ_{r, j}^{'}}{c_{j}^{'}} + a_{r, j}^{'}) - \sinh a_{r, j}^{'}] - \frac{H_{j}^{'}}{2 EA} \\ {L_{r} - Δ_{r, j}^{'} + \frac{c_{j}^{'}}{2} [\sinh 2 (\frac{L_{r} - Δ_{r, j}^{'}}{c_{j}^{'}} + a_{r, j}^{'}) - \sinh 2 a_{r, j}^{'}]} \\ = S_{r} \end{matrix}

(21-3)

Based on the force equilibrium in any hanging point, there is the following recurrence relation between $a_{i + 1, j}^{'}$ and $a_{i, j}^{'}$ (Zhang et al., 2018b)

\begin{matrix} a_{i + 1, j}^{'} = asinh [\sinh (\frac{l_{i, j}^{'}}{c_{j}^{'}} + a_{i, j}^{'}) - \frac{P_{i, j}}{H_{j}^{'}}], \\ i = 1, 2, \dots, n \end{matrix}

(22)

Similar to derivation of equation (16), the span length closure condition yields

\sum_{i = 1}^{n + 1} l_{i, j}^{'} = L + Δ_{l, j}^{'} + Δ_{r, j}^{'}

(23)

The GRG nonlinear solver is applied to the system of equations (20), (21), and (23). The required n+7 unknown parameters are derived, and the pushing distances d_l,j and d_r,j for the left and right saddles, respectively, are assessed as follows

d_{l, j} = Δ_{l, j} - Δ_{l, j}^{'}

(24-1)

d_{r, j} = Δ_{r, j} - Δ_{r, j}^{'}

(24-2)

The proposed algorithm application to a particular engineering project

The analytical algorithm viability was proved by its application to the particular case, namely, the Jindong Bridge, which is a suspension bridge with the longest span across the Jinsha River in Yunnan Province of China (Zhang et al., 2018a). The bridge is spanned as 240 m + 730 m + 120 m, as seen in Figure 7.

Figure 7.

The overall layout of the Jindong Bridge (Unit: m).

A steel truss girder consisting of the main truss, top and bottom bracings, and transverse trusses is used in the bridge construction, as well as prefabricated reinforced concrete (RC) slab decks, which are shown in Figure 8. The main truss is a Warren truss with 5 m height, 17.5 m width, and 5.0 m panel length, and it has a transverse truss in each panel. The bridge floor system combines longitudinal I-beams and a concrete slab deck. The transverse spacing between longitudinal beams is 2.0 m, while beam heights vary from 0.4 to 0.56 m. They are simply supported by the top chords of the cross-members in the main truss. The bridge deck is made from 4.96 m-long, 1.70 m-wide, and 0.16 m-thick precast concrete slabs. The slab deck and longitudinal beams are fastened by shear pins (Zhang et al., 2018a).

Figure 8.

Steel truss girder and bridge floor system of the Jindong Bridge, China: (a) 3D view (Zhang et al., 2018a); (b) cross-section (Unit: mm).

The stiffening girders and bridge floor system of the Jindong Bridge were erected in the following order: (1) the steel truss girder was hoisted, (2) longitudinalI-beams were installed, (3) RC deck slabs were installed, and (4) finally, the pavement and auxiliary facilities were constructed. The permanent load of each part is shown in Table 1.

Table 1.

Permanent load of the stiffening girders and bridge floor system.

Component	Steel truss girder	Longitudinal I-beam	RC deck slab	Deck pavement and auxiliary facilities	Total
Permanent load (kN/m)	45	9	95.5	27.5	177

RC: reinforced concrete.

The steel truss girder of the Jindong Bridge consists of 73 segments, as shown in Figure 9. Cable cranes were used to hoist girder segments. Starting from segment 36, segments were installed symmetrically from the midspan point to both girder-ends (one segment each time). Segments were connected by temporary hinges. During the installation of RC deck slabs, temporary hinges of girder segments were replaced by permanent rigid connections at appropriate instances. Half-deck slabs were installed symmetrically from both girder-ends to the midspan point to generate a channel, while the remaining deck slabs were installed in the opposite direction (Zhang et al., 2018a), as shown in Figure 10.

Figure 9.

Enumeration of the truss segments in the Jindong Bridge, China.

Figure 10.

Installation of concrete deck slabs in the Jindong Bridge, China: (a) the channel; (b) finished installation.

The input parameters required for the analysis are listed in Table 2. The calculated hanger tensile forces of the completed bridge are shown in Figure 11(a). The tensile forces for Nos. 37–71 are omitted for brevity, insofar as they are symmetric to those of hangers Nos. 1–36.

Table 2.

Input parameters of the Jindong Bridge.

Parameter				Symbol	Unit	Value
Span length	Left-side span			L_l	m	240
	Central span			L	m	730
	Right-side span			L_r	m	120
Elevation in thegeodetic system	Point A			h _A	m	860
	Point B			h _B	m	929
	Point M			h _M	m	856
	Point C			h _C	m	929
	Point D			h _D	m	880
Main cable	Weight per unit length		Completed bridge	q	kN/m	14.268
	Weight per unit length		During the construction	q ₀	kN/m	13.805
	Elastic modulus			E	GPa	197.03
	Cross-sectional area			A	m²	0.1759
	Number of the cablesegments betweenpoints B and M			m	/	36
Hanger	Number			n	/	71
	Spacing			l ₁	m	15
				l ₂ –l ₇₁	m	10
				l ₇₂	m	15
Pylon	Bottomcross-section	Area	Left pylon	A_l	m²	18.09
		Area	Right pylon	A_r	m²	18.03
		Section modulus in bending	Left pylon	W_l	m³	36.55
		Section modulus in bending	Right pylon	W_r	m³	36.27
	Height	Left pylon		h_l	m	126
	Height	Right pylon		h_r	m	124
	Weight	Left pylon		G_l	kN	59930
	Weight	Right pylon		G_r	kN	59410
	Flexibility coefficient	Left pylon		ξ_l	m/kN	0.00014864
	Flexibility coefficient	Right pylon		ξ_r	m/kN	0.00014269
	Compressivestrength limit			σ _limit	MPa	22.4

Figure 11.

Hanger tensile forces (a) and unstrained cable lengths (b) in the left half of the central span.

Using the method described in the “Unstrained length of the main cable” section, the unstrained lengths of the main cable in the left- and right-side spans were derived as S_l = 249.028 m and S_r = 129.239 m, respectively. The total unstrained length of the main cable in the central span is 746.956 m. The unstrained length values of each catenary segment in the left half of the central span are shown in Figure 11(b). The right half of the central span is symmetric with the left one, and its data are omitted for brevity. Using the method described in the “Initial pre-offset of the saddle” section, the initial pre-offsets were calculated for the left and right pylon saddles as Δ_l,0 = 1.155 m and Δ_r,0 = 0.416 m, respectively.

The timing and distances of the pylon saddle pushing for the Jindong Bridge for the particular construction stages were calculated by the procedures described in the “Pushing timing of pylon saddle” section and the “Pushing distance of pylon saddle” section, respectively, and listed in Table 3. The total number of five pushings was applied. For the first four pushings, Table 3 shows the current construction step (a) and the next construction step (b) corresponding to each pushing. For the construction step (a), no tensile stresses occurred at the bottom of either left or right pylon, and the maximum compressive stress did not exceed the compressive strength limit. For the construction step (b), the maximum stresses at the bottom of the left and right pylons were smaller than the compressive strength limit. No tensile stresses were observed at the bottom of the right pylon, in contrast to the bottom of the left one. Therefore, it was necessary to push the pylon saddle after the current construction step (a). The last pushing occurs after the construction is finished. Before the last pushing, the compressive strength limit was not reached at the bottom of either left or right pylon.

Table 3.

Calculated timing and distance of pylon saddle pushing.

Pushing No.	If pushing occurs after this construction step	Pylon	Before pushing							Saddle
			Stress at the pylon bottom (MPa)		Pylon top displacement (cm)		Bending moment at pylon bottom
			Min	Max	Calculated	Measured	Caused by horizontal force(MN•m)	Caused by verticalforce(MN•m)	Vertical-to-horizontal ratio	Pushing distance(cm)	Residualpre-offset (cm)
1	(a)*	Left	0.38	7.64	17.1	15.8	145.0	12.4	8.6%	22.7	92.8
	(a)*	Right	3.30	4.86	3.9	4.3	33.6	5.3	15.9%	4.1	37.6
	(b)*	Left	-0.02	8.07	18.9	/	160.2	12.4	7.7%	/	/
	(b)*	Right	3.22	4.97	4.3	/	37.1	5.4	14.5%	/	/
2	(a)*	Left	0.16	8.27	18.9	17.5	160.3	12.1	7.5%	21.8	71.0
	(a)*	Right	3.33	5.33	4.9	4.5	42.3	6.1	14.4%	5.1	32.5
	(b)*	Left	-0.02	8.47	19.7	/	167.3	12.1	7.2%	/	/
	(b)*	Right	3.31	5.39	5.1	/	43.9	6.2	14.1%	/	/
3	(a)*	Left	0.02	9.95	23.1	22.0	196.1	14.5	7.4%	24.8	46.1
	(a)*	Right	3.38	7.03	8.6	8.1	74.5	8.2	11.1%	8.8	23.6
	(b)*	Left	-0.03	10.01	23.3	/	197.9	14.4	7.3%	/	/
	(b)*	Right	3.38	7.07	8.7	/	75.3	8.3	11.0%	/	/
4	(a)*	Left	0.04	11.97	26.8	26.0	227.4	9.4	4.1%	28.3	17.8
	(a)*	Right	3.34	9.54	13.6	12.9	118.2	5.7	4.8%	14.0	9.6
	(b)*	Left	-0.02	12.05	27.1	/	229.9	9.3	4.0%	/	/
	(b)*	Right	3.31	9.59	13.8	/	119.5	5.6	4.7%	/	/
5	Constructionis finished	Left	3.07	10.78	16.7	15.9	141.7	0.7	0.5%	17.8	0
5	Constructionis finished	Right	5.31	9.71	9.2	8.8	80.0	0.3	0.4%	9.6	0

Note: 1-(a): Girder segments Nos. 32-39 are hoisted;

1-(b): Girder segments Nos. 32-40 are hoisted;

2-(a): Girder segments Nos. 24-47 are hoisted;

2-(b): Girder segments Nos. 24-48 are hoisted;

3-(a): The steel truss girder was hoisted and the longitudinal I-beams were installed; the length of the channel formed by the RC deck slabs is 100 m on the left side and 90 m on the right side (as shown in Figure 12(a));

3-(b): The steel truss girder was hoisted and the longitudinal I-beams were installed; the length of the channel formed by the RC deck slabs is 100 m on the left side and 100 m on the right side;

4-(a): The channel was formed by the RC deck slabs. The length of RC deck slabs installed from the midspan to the two girder-ends is 270 m (as shown in Figure 12(b));

4-(b): The channel was formed by the RC deck slabs. The length of RC deck slabs installed from the midspan to the two girder-ends is 280 m.

Figure 12.

The third (a) and fourth (b) pylon saddle pushing moments (unit: m).

The bending moment at the pylon bottom can be split into two constituents induced by the horizontal and vertical forces, respectively. As seen in Table 3, the vertical-to-horizontal ratio of the left pylon is less than 9%, while this ratio of the right pylon is less than 16%. Moreover, this ratio is shown to gradually drop in the construction process.

Before each pushing, the calculated and measured values of pylon top displacement were compared, as shown in Table 3, with their good consistency being revealed. Before the first four pushings of construction step (a), the displacement of the left pylon top was considered as the critical displacement. If this value is exceeded, the left pylon will be unsafe. As seen in Table 3, the critical displacements of the left pylon top (namely, 17.1, 18.9, 23.1, and 26.8 cm) gradually increase as the construction proceeds. This can be attributed to the fact that the downward vertical force exerted by the main cable on the pylon gradually increases in the construction process, which inhibits the occurrence of tensile stresses at the pylon bottom. The internal force and stress at the left pylon bottom are listed in Table 4. As can be seen from Table 4, the total vertical force and the stress produced by it gradually increase in the construction process, due to the increase of the vertical force exerted by the main cable. As a result, the left pylon bottom can bear more and more moment before the occurrence of tensile stresses. Since the moment produced by the unbalanced horizontal force of the main cable is dominant, the critical displacements of the left pylon top gradually increase as the construction proceeds.

Table 4.

Internal force and stress at the left pylon bottom.

Internal force or stress	Unit	Construction step
Internal force or stress	Unit	1-(a)	2-(a)	3-(a)	4-(a)
Vertical force exerted by the main cable, F_{l, j}	MN	12.6	16.4	30.3	48.6
Total vertical force, F_{l, j} + G_l	MN	72.5	76.3	90.2	108.6
Stress produced by the F_{l, j} + G_l	MPa	4.01	4.22	4.99	6.00
Moment, (H_j – H_{l, j})h_l – F_{l, j}(Δ_{l, j} – δ_{l, j})	MN•m	132.6	148.2	181.6	218.0
Stress produced by the moment	MPa	± 3.63	± 4.06	± 4.97	± 5.97

The hanger tensile forces for each construction step listed in Table 3 are summarized in Table 5, as well as the horizontal forces of the main cable in the left-side, central, and right-side spans, and respective catenary parameters. It can be seen that the horizontal forces of the main cable in each span increase as the construction proceeds.

Table 5.

Hanger tensile forces, horizontal forces of main cable, and catenary parameters for each key construction step.

Construction step	Hanger tensile forces (kN)	Horizontal force of main cable (kN)			Catenary parameter
Construction step	Hanger tensile forces (kN)	H_l	H	H_r	a_l	a ₁	a_r
1-(a)	P₃₂–P₃₉ = 225	16643.4	17794.5	17523.6	0.38349	0.44593	0.33490
1-(b)	P₃₂–P₄₀ = 225	16980.9	18252.7	17953.8	0.38152	0.44480	0.33196
2-(a)	P₂₄–P₄₇ = 225	23391.9	24664.4	24323.2	0.35490	0.43268	0.31497
2-(b)	P₂₄–P₄₈ = 225	23734.9	25062.4	24708.4	0.35388	0.43215	0.31296
3-(a)	P₁ = 635.94, P₂–P₉ = 508.75, P₁₀–P₆₃ = 270, P₆₄–P₇₀ = 508.75, P₇₁ = 635.94	40049.6	41605.8	41005.1	0.32543	0.41865	0.39787
3-(b)	P₁ = 635.94, P₂–P₉ = 508.75, P₁₀–P₆₂ = 270, P₆₃–P₇₀ = 508.75, P₇₁ = 635.94	40240.0	41810.5	41203.2	0.32524	0.41855	0.39667
4-(a)	P₁ = 635.94, P₂–P₂₂ = 508.75, P₂₃–P₄₉ = 747.5, P₅₀–P₇₀ = 508.75, P₇₁ = 635.94	70676.7	72481.7	71528.7	0.30730	0.40964	0.35893
4-(b)	P₁ = 635.94, P₂–P₂₁ = 508.75, P₂₂–P₄₉ = 747.5, P₅₀–P₇₀ = 508.75, P₇₁ = 635.94	71022.4	72847.0	71882.9	0.30718	0.40958	0.35935
5	See Figure 11(a)	92786.2	93910.6	93265.4	0.30153	0.40665	0.38432

Noteworthy is that the calculated time schedules and distances of the pylon saddle pushing were successfully applied during the Jindong Bridge construction. This bridge has already passed the completion acceptance by the authorities and it will be opened to traffic very soon.

Discussion

In the algorithm proposed in this article, the geometrical parameters and internal forces at every construction step are calculated based on the conditions of elevation difference closure and the conservation of unstrained length of the main cable segments. The changing process between two adjacent construction steps is effectively avoided in the algorithm of this article, in contrast to FEM. When using FEM, the bridge state of current construction step has to be determined based on the previous one and the changing process between them. In the calculation of the changing process, the geometric nonlinearity induced by the large displacement and the high initial internal force should be taken into account.

When modeling a suspension bridge using FEM, the geometry and initial stress of the main cable are not known at the same time. As a result, repeated forward and backward analyses are needed to determine the bridge states at all construction steps. Fortunately, this drawback of FEM has been overcome by the algorithm proposed in this article. As can be seen from aforementioned sections, repeated forward and backward analyses are effectively avoided.

It should be noted that the vertical force exerted by the girder on the pylon is not taken into account in the calculation of bottom stress. The maximum value of the vertical force exerted by the girder on each pylon column is 486 kN, which is about 0.8% of the weight of each pylon column. If the weight of each pylon column is superimposed with the vertical force exerted by the main cable, the proportion of the vertical force exerted by the girder is much less. Therefore, the vertical force exerted by the girder on the tower can be ignored. On the other hand, since the pushing timing is mainly controlled by the tensile stress at the tower bottom, the calculation result of ignoring the vertical force exerted by the girder is safer.

It is to be noted that the saddle arc and the change of the tangential point between the saddle and the main cable are not taken into account in the calculation of the pushing timing and distance. In order to check the error, two cases are compared in Table 6. In case A, the saddle arc and the tangential point are not considered, in contrast to case B. As can be seen in Table 6, the results of case A show good consistency with those of case B, which indicates the accuracy of the algorithm proposed in this article.

Table 6.

Effects of the saddle arc and the tangential point on the saddle pushing.

Case	Initial pre-offset (m)		Stress at the pylon bottom (MPa)				Pushing distance (m)
			Construction step 1-(a)		Construction step 1-(b)
	Left pylon	Right pylon	Min	Max	Min	Max	Left pylon	Right pylon
A	1.155	0.416	0.381	7.639	-0.025	8.067	22.7	4.1
B	1.141	0.412	0.401	7.614	-0.015	8.052	22.3	4.0

Case A: Saddle arc and tangential point are not considered; Case B: Saddle arc and tangential point are considered.

Conclusions

An analytical algorithm for the assessment of time schedules and distances of the pylon saddle pushing during the construction of a suspension bridge is proposed based on the conditions of (1) conservation of unstrained length of the main cable and (2) closure of elevation difference. The saddle pushing time is assessed by the current stress state analysis at the pylon bottom, which should imply only compressive stresses not exceeding the material compressive strength limit. In this study, the proposed algorithm was successfully applied to a particular suspension bridge under construction with the main span of 730 m. Based on the results obtained, the following conclusions can be drawn.

1.Before each pushing, the unbalanced horizontal forces of the main cable on the two sides of the pylon and the vertical force exerted by the main cable on the pylon contribute to the bending moment at the pylon bottom. The bending moment share caused by the horizontal forces is dominant, while the vertical-to-horizontal moment ratio gradually drops in the construction process.

2. For a suspension bridge with unequal side-span lengths, the pylon on the side with a larger side span is more likely to have tensile stress. This factor controls the most appropriate time of the pylon saddle pushing. Besides, the critical pylon top displacement on the side with a larger side span gradually increases as the construction proceeds. This can be attributed to the fact that the downward vertical force exerted by the main cable on the pylon gradually increases in the bridge erection process, which inhibits the occurrence of tensile stresses at the pylon bottom.

3.The analytical algorithm proposed in this study provided effective calculation of the times and distances of the pylon saddle pushing, is quite straightforward, and its wide application is recommended.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research described in this article was financially supported by the NSFC under the Grant 51678148, a project supported by the Natural Science Foundation of Jiangsu Province (BK20181277), and the National Key R&D Program of China (No. 2017YFC0806009), which are gratefully acknowledged.

ORCID iD

Wen-ming Zhang

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