Synchronous Multi-Span Closure Techniques in Continuous Rigid-Frame Bridges: Research and Implementation

Abstract

This study investigates the Huangdong Daning River Bridge project in Guangxi, where the innovative side-span and mid-span synchronous closure technology for continuous rigid-frame bridges (CRFB) was systematically implemented for the first time in this region of China. A comparative finite element model developed in MIDAS Civil 2024 was employed to analyze the mechanical behavior mechanisms of main girders under span-by-span closure and synchronous closure processes. The numerical simulation results demonstrate that the stress distribution in main girders shows no significant sensitivity (<3%) to closure method differences during both the bridge completion phase and 10-year shrinkage-creep cycle. However, distinct closure sequences (asynchronous vs. synchronous) exhibited notable impacts on the girder alignment at the completion stage. The cumulative deviation induced by differential installation elevations of formwork segments necessitates precise dynamic control during construction monitoring. Furthermore, shrinkage and creep effects manifested differential influences on pier top horizontal displacements and bending moments when employing different closure methods, though all variations remained within 5%. The synchronous multi-span closure technology effectively mitigates structural mutation risks during construction while achieving superior alignment accuracy, rational stress distribution, and accelerated construction progress as verified by field implementation.

1. Introduction

In the field of transportation infrastructure, long-span prestressed concrete continuous rigid-frame bridges are highly regarded for their exceptional spanning capacity, minimal support requirements, high traffic flow efficiency, and esthetic appeal [1,2,3,4,5]. Nevertheless, the segmental construction of such large-span continuous rigid-frame bridge structures generally entails a prolonged and intricate construction process. According to conventional closure techniques, the multi-span closure process requires multiple structural system transformations. As construction progresses, both the structural configuration of the bridge and its load-bearing mechanisms undergo continuous evolution. The final dead-load internal forces within the structure are intrinsically linked to the closure sequence. Distinct construction sequences, characterized by varying initial dead-load internal forces, exhibit differential magnitudes of creep-induced internal force redistribution during structural system transitions. Furthermore, the adoption of different closure sequences significantly impacts both the construction schedule and project costs. Therefore, the selection of an appropriate closure sequence is paramount.

The first long-span prestressed concrete continuous rigid-frame bridge was the Bendendorf Bridge, constructed in 1964 in what was then West Germany, with a main span of 208 m. This bridge utilized thin-walled piers in place of the bulky T-shaped piers typical of rigid-frame bridges, with the side spans formed as a continuous system and the central span equipped with a shear hinge. This bridge type represented the prototype of the continuous rigid frame, maintaining the primary load-bearing characteristics of the T-shaped rigid-frame bridge. The Bendendorf Bridge not only exemplified the advantages of the cantilever construction method but also broke new ground in structural form with the pier-beam consolidation, creating a continuous rigid-frame system with hinges [6]. Subsequently, as high-grade highways demanded smoother and more comfortable driving conditions, the T-shaped rigid-frame bridges with multiple expansion joints could no longer adequately meet the requirements, leading to significant development in the long-span continuous rigid-frame system, which began to be widely applied globally. With further advancements in building materials and construction methods, the Hamanako Bridge in Japan, with a main span of 240 m, was constructed in the 1970s [7]; and in 1985, Australia built the then world’s longest continuous rigid-frame bridge, the Gateway Bridge, with spans of 145 m + 260 m + 145 m [8]. In China, the construction of long-span continuous rigid-frame bridges started later, with the introduction of continuous rigid-frame bridge design and construction from abroad in 1988. In 1990, China completed its first long-span continuous rigid-frame bridge, the Guangdong Luoxi Bridge, with a main span of 180 m [9]. The Luoxi Bridge was the first in China to employ a high-tonnage prestressing system, marking a milestone in the development of rigid-frame bridges in the country. Since then, continuous rigid-frame bridges in China have entered a phase of rapid development. To date, China has constructed over 80 continuous rigid-frame bridges with main spans exceeding 200 m and countless others with spans less than 200 m [10].

Currently, the mature construction technology for continuous rigid-frame bridges generally follows the sequence of “symmetrical cantilever casting → side-span closure → central-span closure”. Due to the large span, multiple continuous spans, and high degree of indeterminacy of long-span continuous rigid-frame bridges, the completion of such bridges requires a long and complex process of structural system transformation. For continuous rigid-frame bridges with multiple spans, this construction sequence results in a prolonged construction period and high costs. Consequently, the traditional closure techniques have the drawback of an extended construction cycle. More significantly, the variability in the closure process can have a pronounced impact on the final permanent load internal force distribution of the multi-span continuous systems. After closure, the bridge structure transitions from a statically determinate to a statically indeterminate one, leading to a redistribution of internal forces. Different closure sequences can also result in considerable differences in the redistribution of internal forces due to creep and temperature changes, thereby affecting the overall mechanical performance of the bridge [11,12,13,14]. Therefore, optimizing the bridge closure process, reducing the number of internal force redistributions, and simplifying the calculation of structural internal forces have become important research areas for promoting the efficient and sustainable construction in the bridge engineering field.

Closure is one of the most critical construction steps in the construction of continuous rigid-frame bridges [15]. Li [16] has proposed a novel closure technique based on the equivalent load method, which involves pre-tensioning the tendons in the bottom slab of the box girder to eliminate the need for counterweights during closure. This approach effectively reduces disturbances during the closure process and aims for efficient construction, but it lacks analysis from the perspective of different closure sequences. Weng [17] and others have developed eight closure schemes for a seven-span continuous rigid-frame bridge based on four principles: closing the side spans before the central span, closing the central span before the side spans, alternating the closure of side and central spans, and sequential closure. These schemes provide a comprehensive mechanical basis for the final closure construction plan, but they do not involve research into and an analysis of simultaneous closure schemes for both side and central spans. Yuan Hui and others [18] have conducted research on the simultaneous closure of a four-span continuous rigid-frame bridge, analyzing the internal forces, deflections, and post-construction impacts of sequential and simultaneous closures, demonstrating the feasibility of simultaneous closure. Wu Haishan and colleagues [19] have further proposed a method for the simultaneous casting of the side-span straight section and closure section using hanging baskets from a construction technology perspective. The study shows that this method has a minimal impact on the forces and deflections of the main girder on the central-span side, but it causes significant disturbance to the main girder on the side span and does not consider the long-term stability of the bridge’s alignment.

In light of this, this paper, based on the advantages of the simultaneous closure of side and central spans in continuous rigid-frame bridges, such as controlling deflection, evenly distributing loads, and reducing the number of statically indeterminate members [20,21,22,23,24,25,26,27], and in conjunction with the construction project of the right span of the Huangdong Daning River Bridge, a three-span continuous rigid-frame bridge in Guangxi, addresses the complex internal force conditions caused by low construction efficiency, the impact of Asynchronous Closure on concrete shrinkage and creep, the influence of temperature changes on measurement values, and the variation in temporary loads. This paper proposes a construction scheme for the simultaneous closure of multi-span continuous systems, analyzing the impact on the stress of the main girder, the alignment of the main girder, and the displacement of the pier top under the sequences of step-by-step closure and simultaneous closure after the completion of the bridge and over a ten-year period of shrinkage and creep. This provides a construction scheme for continuous rigid-frame bridges that ensures correct alignment, reasonable stress distribution, and rapid bridge completion.

2. Theoretical Analysis

For long-span continuous rigid-frame bridges, the geometric alignment and stress states of the main girder during closure construction under different procedures are influenced by cumulative factors from prior construction stages, such as concrete shrinkage, creep, and temperature variations. These factors lead to discrepancies between as-built outcomes and theoretical predictions, resulting in construction deviations. The theoretical analysis aims to establish quantitative relationships between critical parameters (e.g., creep effects) and structural responses of the bridge system. By formulating these dependencies, we identify dominant factors governing the deflection behavior of the two closure methods (synchronous vs. asynchronous). The derived relationships serve as predictive tools to provide a theoretical foundation for deflection calculations in similar large-span bridge projects.

2.1. Theoretical Analysis of Deflection Calculation Under Asynchronous Closure

The theoretical framework of this study is grounded in Chapter 6 of the first edition of High-Pier Long-Span Continuous Rigid-Frame Bridges [28]. In staged construction, the proportional coefficient $v$ formed at point $k$ is a time function related to the loading age $\psi$, as follows:

$$\psi \left(t,t_j,t_0\right)={\psi }_k\left[k+\left(1-k\right)e^{-v\left(t_j-t_0\right)}\right]\left(1-e^{-v\left(t-t_j\right)}\right)$$

(1)

In Equation (1), $t_j$ represents the casting moment of segment $J$ and $t_0$ indicates that section 0 has just started pouring concrete.

To address temporal discrepancies in staged construction, a unified time coordinate system is implemented. This approach resolves ambiguities arising from discrepancies in load application ages, ensuring consistency between theoretical loading timelines and actual field conditions—a prerequisite for accurate long-term performance evaluation. The creep-induced deflection, accounting for time-dependent material behavior, is calculated as follows:

$$\zeta \left(t,T\right)=H_0\left\{1+\alpha \Delta T-{\displaystyle \frac{2}{E{B}_0}}{\displaystyle \sum _{i=L}^j{Q}_i}\left[1+\psi \left(t,t_i,t_0\right)\right]\right\}$$

(2)

In Equation (2), $H_0$ represents the relative height; $\Delta T$ represents the temperature difference between the two measurements before and after tensioning; $E$ and $B_0$ represent the elastic modulus and cross-sectional area of the pier and abutment, respectively; $Q_i$ represents the self-weight of the large beam of segment $i$; $t_i$ represents the time of the completion of segment $i$; and $t_0$ indicates that section 0 has just started pouring concrete.

The deflection under self-weight follows similar rules to creep and compression, and the calculation of deflection due to self-weight is as follows:

$${\beta }_{kj}\left(t\right)=-{\displaystyle \sum _{L=1}^k{R}_{kL}{\displaystyle \sum _{i=0}^j{l}_{L,i}{Q}_{i}1\left[1+\psi \left(t,t_i,t_0\right)\right]}}-S_L{\displaystyle \sum _{i=L}^j{Q}_i}\left[1+\psi \left(t,t_i,t_L\right)\right]$$

(3)

In Equation (3), $R_{kL}={\displaystyle \frac{x_k-{\lambda }_L}{E{I}_L}}dL$, $S_L={\displaystyle \frac{dL}{G{A}_L}}\theta$, $l_{Li}$ represent the equivalent force arm from segment $i$ to the center of segment $L$; $Q_i$ represents the self-weight of the large beam of segment $i$; $t_i$ represents the time of the completion of segment $i$; $t_L$ represents the casting moment of segment $L$; $x_k$ represents the abscissa of point $i$; $x_k$ represents the coordinate of the midpoint of segment $K$; $E$ and $I_L$ represent the elastic modulus and moment of inertia of the segment, respectively; $G$ and $A_L$ represent the shear modulus and cross-sectional area of the segment, respectively; and $d_L$ denotes the length of segment $L$.

The deflection caused by the prestressing tension is expressed as follows:

$${\eta }_{kj}\left(t\right)={\displaystyle \sum _{L=1}^k{R}_{kL}{e}_L{\displaystyle \sum _{i=L}^j{P}_{Lj}}{e}^{-\psi \left(t,t_j,t_L\right)}}\left[1+\psi \left(t,t_i^{\prime },t_L\right)\right]$$

(4)

In Equation (4), $P_{Li}$ denotes the tension force of cable $i$ at cross-section $L$ and $t_i^{\prime }$ denotes the tensioning time at segment $i$.

Under asynchronous closure, the deflection at point $K$ before and after the tensioning of segment $J$ is calculated as follows:

$${\xi }_{kj}^{\varphi }={\eta }_{kj}^i+{\beta }_{kj}^i+{\zeta }_i^{i_j{t}_j}$$

(5)

In summary, the process of asynchronous closure involves complex internal force changes. Due to variations in measurement timing and environmental conditions, all of which affect deflection, the influence of concrete shrinkage and creep is particularly significant. These factors can cause the main girder to gradually deflect downwards, impacting the long-term alignment of the bridge. Additionally, temperature changes induce thermal expansion and contraction effects in the main girder, thereby affecting the accuracy of deflection measurements. Moreover, fluctuations in temporary construction loads also influence the distribution of internal forces. These interwoven factors complicate the calculation and measurement of deflection, necessitating a comprehensive consideration to ensure the structural safety and performance of the bridge.

2.2. Theoretical Analysis of Deflection Calculation Under Synchronous Closure

The theoretical analysis of deflection calculation under the synchronous closure of side and middle spans of a bridge primarily considers the structural forces and deformations. By employing the finite element method, the bridge is discretized into multiple units, and the deflection is determined by solving the equilibrium equations of mechanics.

Based on the analysis of individual beam elements, the rotation angle of the cantilever beam in a continuous rigid-frame bridge is

$$\theta \left(x\right)={\displaystyle {\int }_0^x\frac{M\left(\lambda \right)}{E\left(\lambda \right)I\left(\lambda \right)}}d\lambda +{\displaystyle \frac{T\left(x\right)}{G\left(x\right)A\left(x\right)}}$$

(6)

The deflection at point $K$ is calculated as follows:

$$Y_K\approx Y\left(x_K\right)={\displaystyle \underset{x=0}{\overset{x=x_K}{\int }}{\displaystyle \underset{\lambda =0}{\overset{\lambda =x}{\int }}\frac{M\left(\lambda \right)}{E\left(\lambda \right)I\left(\lambda \right)}}}d\lambda dx+{\displaystyle \underset{x=0}{\overset{x=x_K}{\int }}\frac{T\left(x\right)}{G\left(x\right)A\left(x\right)}}dx$$

(7)

The double integral can be interpreted as an area integral over a triangular region bounded by three vertical lines: $x=\lambda ,x=x,\lambda =0$. By changing the order of integration and integrating with respect to $x$ first, the expression becomes

$${\displaystyle \underset{\lambda =0}{\overset{\lambda =x_k}{\int }}\frac{M\left(\lambda \right)}{E\left(\lambda \right)I\left(\lambda \right)}}\left[{\displaystyle \underset{x=\lambda }{\overset{x=x_k}{\int }}dx}\right]d\lambda ={\displaystyle {\int }_0^{x_k}\frac{M\left(\lambda \right)}{E\left(\lambda \right)I\left(\lambda \right)}}(x_k-\lambda )d\lambda$$

(8)

To numerically approximate the integral over the interval $\left[0,x_k\right]$, the domain is partitioned into K subintervals. Within each subinterval, the integrand is treated as a constant (evaluated at the midpoint value), thereby reducing the integral to a summation:

$$\begin{array}{l}Y\left(x_K\right)={\displaystyle \sum _{L=1}^k{\displaystyle {\int }_{L-1}^L\frac{M\left(\lambda \right)}{E\left(\lambda \right)I\left(\lambda \right)}}}\left(x_k-\lambda \right)d\lambda +\\ {\displaystyle {\int }_{L-1}^L\frac{T\left(x\right)}{G\left(x\right)A\left(x\right)}dx=}{\displaystyle \sum _{L=1}^k{R}_{kL}}M\left({\lambda }_L\right)+S_{L}T\left({\lambda }_L\right)\end{array}$$

(9)

In Equation (9), $M\left({\lambda }_L\right)$ represents the bending moment at the midpoint of segment $L$ after the completion of segment $J$ and $T\left({\lambda }_L\right)$ denotes the torsional moment at the midpoint of segment $L$ following the construction of segment $J$.

$$M\left({\lambda }_L\right)={\displaystyle \sum _{i=L}^{j-1}{P}_{Li}{e}_{\mathrm{L}}1}\left(t-t_0\right)-{\displaystyle \sum _{i=L}^j{Q}_{Li}{l}_{Li}}1\left(t-t_i\right)$$

(10)

$$\mathrm{T}\left({\lambda }_L\right)={\displaystyle \sum _{i=L}^j{Q}_i}1\left(t-t_i\right)$$

(11)

We can represent the moment of elastic deformation using a step function:

$$1\left(t-t_i\right)=\left\{\begin{array}{ll}0& t<t_i\\ 1& t\ge t_i\end{array}\right.$$

(12)

After the completion of section A, the height at Point B is

$$\begin{gathered} H_{kj}^0={\displaystyle \sum _{L=1}^k{R}_{kL}\left[{\displaystyle \sum _{i=L}^{j-1}{P}_{Li}{e}_{L}1\left(t-t_i^{\prime }\right)}-{\displaystyle \sum _{i=L}^j{Q}_i{l}_{Li}1\left(t-t_i\right)}\right]}+S_L{\displaystyle \sum _{i=L}^j{Q}_{i}1\left(t-t_i\right)} \\ +H_0-{\displaystyle \frac{2H_0}{E{B}_0}}{\displaystyle \sum _{i=0}^j{Q}_i{l}_{Li}}1\left(t-t_i\right) \end{gathered}$$

(13)

In Equation (13), $t_i$ represents the completion moment of section; $t_i^{\prime }$ denotes the tensioning moment of section $i$; and $E$ and $B_0$ respectively indicate the elastic modulus of the pier and the cross-sectional area.

Under the condition of the synchronous closure of side and middle spans, the height at point $K$ is calculated as follows:

$$\begin{gathered} H_{kj}^{\varphi }={\displaystyle \sum _{L=1}^k{R}_{kL}\left[{\displaystyle \sum _{i=L}^j{P}_{Li}{e}_{L}1\left(t-t_i^{\prime }\right)}-{\displaystyle \sum _{i=L}^j{Q}_i{l}_{Li}1\left(t-t_i\right)}\right]}+S_L{\displaystyle \sum _{i=L}^j{Q}_{i}1\left(t-t_i\right)} \\ +H_0-{\displaystyle \frac{2H_0}{E{B}_0}}{\displaystyle \sum _{i=0}^j{Q}_i}1\left(t-t_i\right) \end{gathered}$$

(14)

By combining Equations (13) and (14), Equation (15) can be derived:

$$H_{kj}^{\varphi }=H_{kj}^0+{\displaystyle \sum _{L=1}^k{R}_{kL}{P}_{Lj}{e}_{L}1\left(t-t_j^{\prime }\right)}$$

(15)

Although Section 2.1 and Section 2.2 both address theoretical deflection calculations, they diverge fundamentally in their treatment of load types and temporal scales. Section 2.1 employs elastic analytical methods for girder deflection analysis, incorporating time-dependent factors such as concrete shrinkage, creep, and tendon tensioning effects. This approach is tailored for the rapid analysis of asynchronous closure construction. In contrast, Section 2.2 simplifies the girder model by considering only instantaneous elastic deformation (time-independent), making it suitable for the rapid evaluation of synchronous closure scenarios.

These distinct assumptions necessitate separate governing equations. This paper theoretically deduces the deflection calculation of the existing synchronous closing technology. By determining the main factors affecting the deflection of the two closing methods (concrete shrinkage and creep, tensile force, etc.), the deduced relationship will be used as a prediction tool to provide a theoretical basis for the deflection calculation of similar long-span bridge projects.

3. Engineering Application

3.1. Project Overview

Taking the Huangdong Daning River Bridge in Guangxi, China, as the engineering context, which is the first to employ synchronous closure construction within the region, the bridge is designed with a total length of 627.00 m. The span arrangement for both the left and right sections is 5 × 40 + (77 + 145 + 77) + 3 × 40. The superstructure of the bridge is of the prestressed continuous rigid-frame type. The main bridge, with a length of 299 m, is configured as a dual-deck bridge, with each deck having a width of 12.75 m. The substructure piers are arranged in separate decks, featuring hollow thin-walled rectangular piers. The piers are designed with rounded corners along the flow direction, and the maximum height of the piers reaches 74.132 m, as illustrated in Figure 1.

The Huangdong Daning River Bridge is designed with separate decks, featuring a total of four main piers located at P6# and P7# for both the left and right decks. The 0# segment has a length of 11 m and a beam height of 8.7 m (measured at the center of the box girder). The portion of the bottom slab extending from the pier body is 8 m wide, with a cantilevered section extending 7 m outward. The wing plates on both sides have cantilever arms, each measuring 2.875 m. The outer webs are straight, the top slab is 12.75 m wide, and the bottom slab is 1.26 m thick, with cantilevers extending 2.5 m along the bridge’s longitudinal direction on both sides. The superstructure of the main bridge is constructed using hanging basket cantilever casting, comprising four side-span cast-in-place sections, four side-span closure segments, and two central-span closure segments.

3.2. Construction Methodology

This bridge comprises three closure segments. The construction processes for synchronous and asynchronous closure are detailed as follows:

Synchronous Closure: All closure segments are constructed in parallel. The workflow includes the following: preparation work → formwork installation, reinforcement bar binding, and prestressing tendon placement → rigid-frame closure locking (conducted at optimal temperatures) → synchronous concrete pouring for both side-span and mid-span closure segments → simultaneous prestressing tendon tensioning.

Non-Synchronous Closure: The sequential workflow involves the following: preparation work → formwork installation, reinforcement bar binding, and prestressing tendon placement → rigid-frame closure locking (performed under appropriate temperature conditions) → concrete pouring for the side-span closure segment → the tensioning of side-span prestressing tendons → concrete pouring for the mid-span closure segment → the tensioning of mid-span prestressing tendons. Asynchronous closure typically involves first completing the side-span closure, followed by the midspan closure, whereas synchronous closure simultaneously executes both side and midspan closures. A schematic comparison between asynchronous and synchronous closure methods is illustrated in Figure 2 below.

3.3. The Establishment of the Finite Element Model

Based on the plane frame theory, the bridge analysis software MIDAS Civil 2024 was employed to model and analyze the two different closure sequences according to the design drawings. It is essential that the structural calculation model and boundary conditions correspond precisely with the actual structure. The CEB-FIP 2010 model was adopted to simulate concrete shrinkage and creep effects. Time-dependent creep compliance was integrated via the age-adjusted effective modulus (AAEM) method, with a user-defined material subroutine (UMAT) embedding nonlinear creep kernel functions. The entire bridge is discretized into 123 nodes and 120 elements, with the superstructure comprising 95 nodes and 94 elements and the substructure consisting of 28 nodes and 26 elements. With the base consolidation of the main pier, the pier and beam positions are rigidly connected with elastic connections. The position of the abutment releases its translational/rotational direction according to its support type. A structural discretization diagram of the Huangdong Daning River Bridge is illustrated in Figure 3.

The term ‘Main Girder Node’ refers to finite element mesh nodes distributed along the longitudinal axis of the girder (spacing: 2.5 m). There are 87 main beam joints, from left to right, and node numbering corresponds to physical bridge locations as shown in Figure 4 (The arrows represent the beam node coordinates from small to large).

4. Comparative Analysis of Different Closure Methods

4.1. Influence of Different Closure Sequences on Main Girder Stress

The calculated stress results for the upper and lower edges of typical main girder sections under both asynchronous and synchronous closure procedures are presented in

Table 1. Stress in the upper edge of the main girder under different closure sequences.

Node Position	Bridge Completion Stage Top Flange Stress (MPa)			Top Flange Stress After 10 Years of Shrinkage and Creep (MPa)
	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①
Left Support	−3.24	−3.25	0.31%	−3.08	−3.07	0.32%
Quarter Span of Side Span	−8.1	−8.23	1.60%	−7.35	−7.42	0.95%
Midspan of Side Span	−8.99	−9.13	1.56%	−8.03	−8.10	0.87%
Three-Quarter Span of Side Span	−10.8	−11.0	1.85%	−9.81	−9.87	0.61%
Quarter Span of Main Span	−10.2	−10.4	1.96%	−9.42	−9.49	0.74%
Midspan of Main Span	−4.79	−4.94	3.13%	−4.80	−4.88	1.67%
Three-Quarter Span of Main Span	−10.2	−10.40	1.96%	−9.43	−9.50	0.74%
Right Support	−3.24	−3.25	0.31%	−3.08	−3.07	0.32%

and

Table 2. Stress at the bottom edge of the main girder under different closure sequences.

Node Position	Bridge Completion Stage Top Flange Stress (MPa)			Top Flange Stress After 10 Years of Shrinkage and Creep (MPa)
	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①
Left Support	−5.64	−5.65	0.18%	−5.34	−5.32	0.37%
Quarter Span of Side Span	−6.59	−6.42	2.60%	−6.78	−6.64	2.07%
Midspan of Side Span	−7.05	−6.89	2.27%	−7.53	−7.43	1.33%
Three-Quarter Span of Side Span	−8.18	−8.06	1.47%	−8.6	−8.53	0.81%
Quarter Span of Main Span	−6.97	−6.88	1.30%	−6.98	−6.94	0.57%
Midspan of Main Span	−7.21	−6.84	5.14%	−5.94	−5.67	4.55%
Three-Quarter Span of Main Span	−6.97	−6.88	1.30%	−6.97	−6.93	0.57%
Right Support	−5.64	−5.65	0.18%	−5.34	−5.32	0.37%

, respectively. The calculated results of the stress at the top edge of all sections of the main girder at the bridge completion stage and after 10 years of shrinkage and creep are shown in Figure 5 and Figure 6, respectively. The calculated results of the stress at the bottom edge of all sections of the main girder at the bridge completion stage and after 10 years of shrinkage and creep are shown in Figure 7 and Figure 8, respectively.

As shown in Table 1 and Table 2 and Figure 3, Figure 4, Figure 5 and Figure 6, during the bridge completion stage, the maximum stress at the top edge of the main girder of the three-span rigid-frame bridge during the sequential closure sequence is −10.8 MPa, while the maximum stress at the same location during the simultaneous closure sequence is −11.0 MPa, with a difference of 0.2 MPa, representing a 1.85% increase for the former over the latter. For the bottom edge of the main girder, the maximum stress during the sequential closure sequence is −8.18 MPa, and during the simultaneous closure sequence, it is −8.06 MPa, with a difference of 0.12 MPa, indicating a 1.47% increase for the former over the latter. Moreover, there is no tensile stress in the cross-section of the main girder. The numerical analysis results indicate that the stress distribution in the main girder under the completed bridge state does not show significant sensitivity to the closure sequence. A further comparison of the stress monitoring data over ten years of operation reveals that the variation in stress at critical sections remains within 3%, confirming that the time-dependent effects of shrinkage and creep on the mechanical behavior of the main girder are not significantly related to the choice of the closure scheme. This mechanical characteristic provides important theoretical support for the optimization of the closure process in multi-span continuous rigid-frame bridges.

4.2. Influence of Different Closure Sequences on the Main Girder’s Linear Shape

Different closure methods result in varying structural stress states upon bridge completion. In hyperstatic structures, the stress state under constant load significantly influences the later shrinkage, the creep of concrete, and the effectiveness of prestressing, thereby affecting the structure’s long-term deflection. A comparative analysis of the vertical displacement of the main girder was conducted for the two different procedures: asynchronous and synchronous closure. The results for typical sections of the main girder are presented in

Table 3. Main girder linear shape under different closure sequences.

Node Position	Vertical Displacement of Main Girder at Completed Bridge Stage (mm)			Vertical Displacement of Main Girder After 10 Years of Shrinkage and Creep (mm)
	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①
Left Support	−8.78	−8.77	0.08%	−8.79	−8.79	0.05%
Quarter Span of Side Span	−39.51	−61.14	−54.74%	−37.06	−58.64	−58.22%
Midspan of Side Span	−13.47	−28.98	−115.16%	−8.44	−23.82	−182.39%
Three-Quarter Span of Side Span	−3.91	−10.34	−164.23%	−0.56	−6.90	−1131.23%
Quarter Span of Main Span	−8.00	5.58	169.70%	−19.16	−6.02	68.58%
Midspan of Main Span	11.23	8.53	24.05%	−8.51	−12.69	−49.10%
Three-Quarter Span of Main Span	−7.97	5.85	173.38%	−19.18	−5.79	69.82%
Right Support	0.35	0.22	36.39%	0.64	0.52	18.34%

. The vertical displacement of all sections of the main girder at the bridge completion stage is shown in Figure 9, and the vertical displacement of all sections after ten years of shrinkage and creep is illustrated in Figure 10.

As illustrated in Table 3 and Figure 9 and Figure 10, during the bridge completion stage, the maximum vertical displacement of the three-span rigid-frame bridge under sequential closure is 11.23 mm, whereas under simultaneous closure, it is 8.53 mm, with both occurrences at the mid-span closure section. The displacement under sequential closure is 24.05% greater than that under simultaneous closure, equating to a difference of 2.70 mm. Furthermore, the computational results reveal that significant changes occur at certain joint positions of the main girder when adopting the two different closure sequences. For example, at the one-quarter point of the side span, the vertical displacement is 39.51 mm downwards under sequential closure compared to 61.14 mm downwards under simultaneous closure, with the former being 54.74% less than the latter, constituting a difference of 21.63 mm. The above analysis demonstrates that the two different closure sequences—sequential and simultaneous—have a significant impact on the main girder’s linear shape during the bridge completion stage. The discrepancy in the installation elevation of the segmental formwork directly leads to the accumulation of positioning deviations, necessitating precise dynamic control during the construction process. Further research indicates that during the ten-year operation period, the cumulative variation amplitude of vertical displacement at key monitoring sections remains stable within a 1.5 mm threshold. The computational data confirm that the structural deformation characteristics caused by the time-dependent effects of concrete shrinkage and creep do not show a significant correlation with the choice of closure technique path. Asynchronous Closure: Sequential loading induces incremental displacements that accumulate geometrically. Synchronous Closure: Full-section loading eliminates inter-stage coupling, yielding displacement dominated by single-phase effects.

4.3. Influence of Different Closure Sequences on Pier Top Displacement and Internal Forces

During the bridge completion and ten-year shrinkage and creep stages, a comparative analysis was also conducted on the displacement and mechanical performance of the piers under two different closure sequences: asynchronous closure and simultaneous closure. The calculated results of the horizontal displacement at the pier top and the internal force moment during the closure stage under different closure construction sequences are presented in

Table 4. Horizontal displacement and bending moment at pier tops during the completed bridge stage.

Node Position	Horizontal Displacement at Pier Tops (mm)			Bending Moment at Pier Tops (kN·m)
	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①
Pier 6	10.31	10.06	2.42%	−23021	−23331	−1.35%
Pier 7	−13.38	−10.58	20.93%	23,458	23,665	−0.88%

and

Table 5. Horizontal displacement and bending moment at pier top after ten years of shrinkage and creep.

Node Position	Horizontal Displacement at Pier Tops (mm)			Bending Moment at Pier Tops (kN·m)
	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①	① Asynchronous Closure	② Synchronous Closure	(① − ②)/①
Pier 6	16.96	17.58	−3.66%	−31,877	−33,212	−4.19%
Pier 7	−17.44	−18.07	−3.61%	32,638	33,895	−3.85%

. The sign conventions are as follows: displacements are considered positive when directed horizontally to the right and negative when to the left; moments are considered positive when they produce a clockwise rotation around the positive y-axis and negative when counterclockwise.

Table 4 and Table 5 illustrate that the pier top displacement of a three-span rigid-frame bridge under sequential closure is greater than that under simultaneous closure, with a maximum difference of 20.93%. The maximum difference in pier top bending moment reaches −1.35%. Further research indicates that shrinkage and creep effects have a certain influence on the changes in pier top horizontal displacement and bending moment when different closure methods are employed, albeit within a 5% range. The maximum difference in pier top displacement is −3.66%, and the maximum difference in pier top bending moment is −4.19%.

5. Comparative Analysis of Field-Measured Results for Synchronous Closure

5.1. Site Arrangement for Synchronous Closure

During the construction process, strict monitoring should be implemented, following a cyclic sequence: theoretical calculation and prediction → phased construction based on the prediction → actual measurement and feedback upon the completion of the phased construction → parameter analysis, assessment, and optimization based on the feedback → theoretical calculation and prediction for the next construction phase.

5.2. Field Measurement

Based on the structural stress characteristics of the Huangdong Daninghe Bridge, the key to its construction control lies in the management of structural alignment and stress. The progressive formation of the continuous rigid-frame bridge’s mechanical system essentially represents a structural system reconstruction process under the coupled effects of multi-stage construction. The stress and vertical alignment of the main girder are highly sensitive to factors such as the precision of prestress tensioning, the cross-sectional dimensions of the girder, the performance and casting weight of the concrete material, the construction period, and structural temperature, all of which should be prioritized for precision control. Under the synchronous closure sequence, the entire closure construction process was tracked in real time, with the stress and alignment results of the synchronous closure monitored. Measurement control points were established in the closure cast-in-place concrete girder segments. Leveling instruments were employed to monitor the quarter-span sections of the main girder, with the sensor layout illustrated in Figure 11 (the symbol ‘▼’ denotes elevation control points). For stress measurements, BGK-4000 series vibrating wire strain gauges (Geokon, Lebanon, NH, USA) were installed at the top of both piers and the mid-span section, as detailed in Figure 11 (the symbol ‘●’ denotes stress control points). The blue numbers on the diagram represent the length of the bridge in cm, and the blue # at the bottom of the pier column represents the number of the piers.

The monitoring results of the main girder stress are presented in

Table 6. Stress table of main beam before and after synchronous closing.

Node Position	Stress at the Top Edge of the Main Girder Before and After Synchronized Closure (MPa)			Stress at the Bottom Edge of the Main Girder Before and After Synchronized Closure (MPa)
	① Measured Value	② Theoretical Value	(① − ②)/①	① Measured Value	② Theoretical Value	(① − ②)/①
Left Quarter Span	−6.01	−6.34	−5.49%	−6.46	−6.25	3.25%
Left Side of Pier 6	−12.54	−12.85	−2.47%	−6.36	−6.18	2.83%
Right Side of Pier 6	−12.49	−12.66	−1.36%	−6.34	−6.09	3.94%
Midspan of Central Span	−3.95	−4.14	−4.81%	−7.84	−8.21	−4.72%
Left Side of Pier 7	−14.66	−15.22	−3.82%	−5.13	−4.84	5.65%
Right Side of Pier 7	−13.96	−14.42	−3.30%	−5.74	−5.65	1.57%
Right Quarter Span	−5.99	−6.33	−5.68%	−6.46	−6.26	3.10%

and Figure 12, while the monitoring results of the main girder alignment are shown in

Table 7. Synchronous closure and main girder alignment before and after.

Node Position	Main Girder Vertical Displacement (mm)
	① Measured Value	② Theoretical Value	(① − ②)
Left Quarter Span	5.00	3.88	1.12
Midspan of Side Span	3.00	0.65	2.35
Three-Quarter Span of Side Span	1.00	0.50	0.50
Quarter Span of Central Span	8.00	9.01	−1.01
Midspan of Central Span	36.00	31.00	5.00
Three-Quarter Span of Central Span	10.00	9.00	1.00

and Figure 13.

Based on the data presented in Table 6 and Table 7 and Figure 12 and Figure 13, after the synchronous closure, the measured values of stress and vertical displacement for each main girder section of the three-span rigid-frame bridge are generally in line with the theoretical values, with errors within the permissible range. Specifically, the maximum difference in main girder stress is only 0.56 MPa, occurring at the left side of pier 7#, and the measured vertical displacement of the main girder section is also consistent with the theoretical values, with a maximum difference of only 5 mm, occurring at the mid-span of the central span. The simultaneous closure of multiple side and central spans can shorten the overall closure construction period, making the process more compact. For statically determinate structures, concrete shrinkage and creep do not induce secondary internal forces, and the calculated deflection values under various conditions are easily consistent with the measured values. However, for statically indeterminate structures, there tends to be some deviations between the calculated and measured values. Therefore, conducting a single closure is highly beneficial for deflection control.

6. Conclusions and Recommendations

(1)

Simultaneous closures of multiple spans can shorten the overall closure duration and streamline the construction sequence. Moreover, when a multi-span continuous system is closed simultaneously, the loads from the closure segments act on the final structure concurrently, resulting in more uniform internal force distribution. This approach simplifies the complex internal force calculations that arise from the sequential closure method, where secondary internal forces increase with the degree of hyperstaticity and continuously alter the structural form.

(2)

As the final and most critical phase in the construction of long-span continuous rigid-frame bridges, the closure sequence—whether asynchronous or synchronous—significantly impacts the geometric alignment of the girder. In this project, the alignment discrepancy reached 54.74%, primarily attributed to cumulative deviations in formwork positioning caused by variations in segmental formwork installation elevations. These findings underscore the necessity for precision-controlled dynamic adjustments during construction to mitigate error propagation.

(3)

Numerical analysis reveals that the stress distribution in the main girder of the completed bridge exhibits low sensitivity (<3%) to variations in closure sequences. A further analysis of decade-long stress monitoring data demonstrates that key cross-sectional stress fluctuations remain within a 3% tolerance threshold. While shrinkage and creep effects under different closure methods influence horizontal displacements and bending moments at pier tops, these variations are confined to a 5% tolerance band. Maximum horizontal displacement difference at pier tops: −3.66%. Maximum bending moment difference at pier tops: −4.19%. These results confirm that both closure methodologies yield structurally robust performance over extended service periods, with time-dependent material effects posing minimal risks to global integrity.

(4)

The synchronous closure technique for multi-span rigid-frame bridges, while innovative and efficient, presents several critical limitations that must be carefully addressed in design and construction. These limitations include the following: Coordinating simultaneous operations across multiple spans demands the rigorous real-time monitoring and synchronization of equipment (e.g., formwork, tensioning jacks). Any delay or error in one span propagates systemic risks, increasing the likelihood of construction defects.

(5)

Through research on the effects of different closure methods on main girder stress and alignment, this paper not only theoretically validates the feasibility of the new approach but also achieves shorter construction durations and higher economic efficiency in practice. It provides a new technological option for the construction of three-span rigid-frame bridges in special environments.