Research on the Control Method for the Reasonable State of Self-Anchored Symmetry Suspension Bridge Stiffening Girders

Abstract

Most existing methods for the determination of the reasonable finished state of self-anchored symmetry suspension bridges were based on the stress state of the stiffening girders used in the construction. A simple and practical control method for the reasonable completion state of stiffened beam based on double control indexes of deformation and stress was proposed. In this paper, the long-term effects of shrinkage and creep were taken into consideration, and a finite element model was built to study the change in the stiffening girder stress during operation. The mid-span deflection of the middle span sustained increasing and the compression stress in the bottom slab of the stiffening girder consistently decreased under the effects of shrinkage and creep. The speed changes from fast to slow and tends to become stable in 50 years. Furthermore, stiffening girders under the action of hanger force, dead weight, cable force, and pre-stress were investigated to study the mechanism of the stress change during operation. Based on the safety stress state of stiffening girders after 50 years, a new control method for the reasonable finished state was proposed. Moreover, the total cross-section of stiffening girders maintained the compression stress state during the developing processes of shrinkage and creep in 50 years. Finally, the utilization in the Hunan Road self-anchored symmetry suspension bridge verified the simplicity and practicality of this new control method and confirms that the method can be implemented to guide the design and construction of the similar bridges.

1. Introduction

The bridge structure has superior structural performance in the state of symmetrical design and construction [1,2,3]. A self-anchored suspension bridge is a common and beautiful symmetrical engineering structure, which has been widely used in the process of urban construction [4,5,6]. Self-anchored symmetry suspension bridges are a type of flexible composite system consisting of a main cable, hanger, concrete stiffening girder, and tower [7]. The construction process, differing from that of earth-anchored symmetry suspension bridges, is initiated by constructing the girder before erecting the cable, then conducting system conversion by tensioning the hanger and finally reaching the finished bridge state [8,9,10,11]. For the self-anchored symmetry suspension bridge, the finished state is of great importance to safe operation in the long term [12]. The effects of shrinkage and creep of the concrete stiffening girder and tower lead to changes in the cable and hanger force, as well as the mechanical properties of the stiffening girder [13]. It is therefore important to take the long-term effects of shrinkage and creep during operation into consideration [14]. Currently, there are considerable research efforts towards hanger force determination and calculation methods for the main cable shape based on the reasonable stress state of the stiffening girder, as well as a few attempts to address the effects of shrink and creep on self-anchored suspension. Li et al. obtained a method for the determination of a reasonable hanger force in view of the mechanical properties of the stiffening girder. Furthermore, a method with high efficiency and high precision for the calculation of the line shape of the main cable and the hanger parameters was developed for the determination of the finished bridge state [15]. Li et al. proposed a systematic and generic procedure for the determination of the reasonable finished state of self-anchored symmetry suspension bridges [16]. Zhou et al. studied the determination method and implementation process of reasonable completion state for the Hunan Road Bridge which is the widest concrete self-anchored suspension bridge in China [17]. Li et al. presented the rigid sustainer continuous beam method, which is suited for hanger force optimization. In addition, based on the fundamental equilibrium differential equations of suspension cables, Li obtained the beam analogy method for the calculation of the line shape of the main cable. The two methods were combined and applied to address the determination of the reasonable finished state of self-anchored symmetry suspension bridges. The veracity and effectiveness were also identified [18]. Qiao et al. studied the impact of shrinkage and creep during construction on the main structural element internal forces and deformation using the non-linear finite element model of a two tower, two cable, three-span self-anchored concrete symmetry suspension bridge. Construction adjustment measures were proposed to address the effects of shrinkage and creep to ensure the reasonableness of the finished state [19]. Hu et al. deduced the finite element expression of shrinkage and creep effects on the structure through the initial strain method, where the age of loading was taken into account. Further study based on the Pingsheng Bridge in Guangdong obtained the consequences of the effects of shrinkage and creep on self-anchored suspension after bridge completion [20].

However, in the above studies, the effects of shrinkage and creep during the long-term operation were not taken into consideration when determining the reasonable finished state of a self-anchored symmetry suspension bridge. A simple and practical control method of reasonable completion state of stiffened beam based on double control indexes of deformation and stress considering the effects of shrinkage and creep during the long-term operation was proposed. In this paper, the analysis method of shrinkage and creep in the Code for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges [21] and culverts and step-by-step finite element analysis method were utilized to study the process and mechanism of the changes in the internal force during long-term operation. Finally, we proposed a control method for the reasonable finished state of self-anchored symmetry suspension bridges, which takes the long-term effects of shrinkage and creep into consideration.

2. Effects of Shrinkage and Creep on Concrete Stiffening Girders

2.1. Description of the Hunan Road Bridge

The Hunan Road Bridge in Liaochen Shandong is a self-anchored concrete symmetry suspension bridge with two-tower, two-main cable plane. The span layout of the 218 m-long bridge is 53 + 112 + 53 m. The main cables of the symmetry suspension bridge have 37 strands, each of which is composed of 169 galvanized parallel steel wires with a diameter of 5.1 mm. The transverse center distance of the cables is 31.7 m, the ratio of the sag-to-span of the main span is 1/5.276, and that of the side span is 1/12.965. There are 37 couples of vertical hangers, with 21 supporting the main span and 8 supporting each side span. The hangers are evenly spaced with a distance of 5 m, each of which consists of 127 high-strength galvanized parallel steel wires with a diameter of 7.1 mm. The devised hanger tension is 3300 kN at a typical segment of the bridge. The stiffening girder is a C50 concrete double-box girder; each of the boxes has triple cells. The height of the standard cross-section is 2.8 m, and the width is 53 m. The prestressed tendons of the stiffening girder are symmetrically arranged in straight lines. Figure 1 shows the general layout of the Hunan Road Bridge.

2.2. Computational Theory of Shrinkage and Creep

Substantial test research efforts have shown that the ultimate creep deformation has a linear relationship with the initial instantaneous elastic deformation when the concrete stress is less than the ultimate strength. The superposition principle applies to the strain generated by stress applied in batches [22,23]. According to the linear computational theory of creep, the equation of the relationship between the stress increment and strain increment of concrete produced by creep in the time interval t_i can be written as:

$$\Delta {\mathit{\epsilon }}_{\mathit{c}}\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)=\frac{\mathbf{\Delta }{\mathit{\sigma }}_{\mathit{c}}\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)}{\mathit{E}\left({\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)}\left[\mathbf{1}+\mathbf{\chi }\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)\mathit{\varphi }\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)\right]+{{\displaystyle \sum }}_{\mathit{j}=\mathbf{1}}^{\mathit{i}-\mathbf{1}}\frac{\mathbf{\Delta }\mathit{\sigma }\left({\mathit{t}}_{\mathit{j}}\right)}{\mathit{E}\left({\mathit{t}}_{\mathit{j}}\right)}\left[\mathit{\varphi }\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{j}}\right)-\mathit{\varphi }\left({\mathit{t}}_{\mathit{i}-\mathbf{1}},{\mathit{t}}_{\mathit{j}}\right)\right]$$

(1)

$${\mathbf{\chi }}_{\left(\mathit{i}, \mathit{i}-\mathbf{1}\right)}=\frac{\mathbf{1}}{\mathbf{1}-{\mathit{e}}^{-\mathit{\varphi }\left(\mathit{i},\mathit{i}-\mathbf{1}\right)}}-\frac{\mathbf{1}}{\mathit{\varphi }\left(\mathit{i},\mathit{i}-\mathbf{1}\right)}$$

(2)

where subscript c was produced by creep; ϕ is the creep coefficient; and χ(i, i−1) is the ageing coefficient, which ranges from 0.6 to 0.9, and is 0.8 in general.

The age-adjusted effective modulus can be defined as:

$${\mathit{E}}_{\mathit{\varphi }}\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)=\frac{\mathit{E}\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)}{\mathbf{1}+\mathit{\chi }\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)\mathit{\varphi }\left({\mathit{t}}_{\mathit{i}},{\mathit{t}}_{\mathit{i}-\mathbf{1}}\right)}$$

(3)

Based on the linear relationship between the stress or internal force increment and the strain increment produced by creep in the time period i, finite element analysis can be used to calculate the internal force and deformation produced by creep in each time interval. First, substitute the age-adjusted effective modulus for the modulus of elasticity of concrete. Then, analyze the effects of creep in the time interval in sequence and accumulate to obtain the final results of each time interval. Figure 2 shows the calculation flowchart for the effects of creep.

The node internal force and displacement increment can be easily determined by a similar finite element method used to analyze the effects of temperature. Thus, the shrinkage strain in i time can be easily analyzed.

2.3. Finite Element Analysis

The 3-D FE model (Figure 3) of the bridge is established to perform the analysis using Midas/Civil. The stiffening girder was simulated by a double main beam, and the C50 concrete parameters of the girder in the model are as follows: γ_c = 26 kN/m³; the parameters of the hangers: γ_s = 78.5 kN/m³, Es = 2.05 × 105 MPa, f_pk = 1670 MPa. Cable elements were used to model the cable and hanger, and the hanger tension was modeled by external force. The prestressed tendons of the stiffening girder are arranged in straight lines in the model, and the parameters are as follows: f_pk = 1860 MPa and σ_con = 1395 MPa. There are 876 nodes and 981 elements in the entire bridge model. The calculation model of FEM analysis adopts the theory of geometric nonlinearity and material nonlinearity. One iteration of the finite element analysis of structures took about 2 to 3 min. The validity of the finite element model can be verified by comparing the theoretical calculation data with the actual bridge monitoring data which was shown in

Table 1. Displacement and stress of bridge in FEM model and actual bridge in service stage.

Items	FEM Model	Actual Bridge	Maximum Error
Cable force Tc/kN	[0.998, 1.021]	1	[−0.2%, +2.1%]
Hanger force TH/kN	[0.999, 1.013]	1	[−0.1%, +1.3%]
Displacement/cm	[0.996, 1.015]	1	[−0.4%, +1.5%]
Effective prestress Npe/MPa	[0.995, 1.009]	1	[−0.5%, +0.9%]
Bending moment /kN·m	[0.999, 1.017]	1	[−0.1%, +1.7%]

.

The construction stage was established in the model based on the practical construction situation. Large deformation geometric nonlinear theory was taken into consideration in the analysis process. The age-adjusted effective modulus method was used to perform the analysis of the effects of shrinkage and creep. The finished bridge state was defined after brackets are removed as the initial completed bridge state, and the bridge state after 50 years of the action of shrinkage and creep as the final completed bridge state. Then, from the initial state to the final completed bridge state, the stress changes under permanent action in stiffening girder’s upper and lower margin were studied.

2.4. Effects of Shrinkage and Creep on the Mechanical Properties of the Stiffening Girders

Figure 4 shows the stress in the stiffening girder’s upper and lower margins of the initial and final completed bridge state. The initial completed bridge state was determined by the optimized rigid supported continuous beam method.

As shown in Figure 4, the total cross-section of the stiffening girders is under a compression state in the initial completed bridge state. The stress in the upper margin of the girder ranges from −5.4 MPa to −7.0 MPa; in the lower margin, the stress ranges from −4.1 MPa to −9.0 MPa.

As time passes, significant changes of stress and displacement are found under the effects of shrinkage and creep. The creep effect makes the deformation increase and the stress uniform. The stress in the girder’s lower margin in the initial and final completed bridge states in Figure 4 shows that in the middle span zone (from the 13th hanger to the 25th hanger) and most of the side span zone (from the 1st hanger to the 6th hanger, as well as from the 32nd hanger to the 37th hanger), the compression stress decreases constantly. The stress of the girder’s lower margin in the mid-span of the middle span changes from −5.7 MPa to 0.3 MPa, which shows that the lower margin of the girder in the mid-span is in tension state at the end, and the range occupies approximately 20% of the middle span. Near the towers (from the 7th hanger to the 12th hanger, and from the 26th hanger to the 31st hanger), the compression stress of the girder’s lower margin keeps increasing. In the sections at hanger 8 and hanger 9 (defined as H8 and H9), which are beside the tower, the compression stress increases by 5 MPa. To the upper margin of the girder, the compression stress increases in the mid-span (from the 13th hanger to the 25th hanger) and decreases in the rest zone of the girder. In the middle span, the stress in the girder’s upper margin of the mid-zone changes from −5.4 MPa to −8.0 MPa. In sections H8 and H9, the stress decreases by 4.3 MPa.

Thus, in the mid-span zone of the middle span, the stress in the lower margin of the girder is changed more than in the rest of the girder. In the final completed bridge state, tensile stress appears in the mid-span zone of the stiffening girder, which is bad for the safety of the concrete stiffening girder. Under the action of variable loads, the tensile stress zone may crack, so it is important to control the stress state of the stiffening girder, especially the lower margin in the mid-span zone, which may be in a tensile stress state.

The typical sections at hanger 4, 12, 15, and 19 are chosen to perform the stress analysis to study the shrinkage and creep effects on the stress in the lower margin of the stiffening girder. As shown in Figure 5, the stress decreases with time. In the first 5 years, the stress reduction in sections H4, H12, H15, and H19 is 85%, 82%, 73%, and 72%, respectively, of the total reduction in 50 years. The stress reduction in each section reaches 90% in 15 years. Then, the stress in sections H4 and H12 tends to be stable at 20 years and no longer decreases. However, the stress in sections H15 and H19 keeps decreasing gradually, and the stress reduction at 40 years reaches 98% of the total reduction. The stress changes caused by shrinkage and creep mostly occur in the first 15 years and tend to be constant at 50 years. Therefore, the stress state in 50 years is set as the target of the final completed bridge state. Furthermore, the mid-zone of the stiffening girder in the middle span should be given greater attention on account of the greater stress changes.

3. Mechanism of Stress Changes

Table 2. Mechanical property changes between the initial and final completed bridge states.

Items	Position	Initial State	Final State
Cable force Tc/kN	Anchor span	52,133	45,695
Hanger force TH/kN	H15	3251	2971
	H19	3223	2953
Displacement/cm	H15	/	−13.8(↓)
	H19	/	−17.3(↓)
Effective prestress Npe/MPa	H19	1168	1074
Bending moment/kN·m	H15	−2515	26,487
	H19	−1291	40,836

shows the mechanical property changes between the initial and final completed bridge states. The cable force reduced by 12%, and the hanger force and the effective prestress both reduced by 8%. The mid-span deflection was 17.3 cm. These changes inevitably lead to severe alterations to the stiffening girder mechanical behavior and affect the structure safety performance.

The concrete stiffening girder of self-anchored symmetry suspension bridge burdens various permanent actions. The main actions are as follows: the effective prestress (N_pe), the cable force (T_c), the hanger force (T_H), and self-weight (G) (including the first and the secondary dead load). Therefore, the normal stress σ_x of the section consists of axial normal stress σ_ax and bending normal stress σ_bx, which are caused by the main actions above. The equation of σ_x can be written as:

$${\sigma }_{\mathit{x}}={\sigma }_{\mathit{ax}}+{\sigma }_{\mathit{bx}}=\left({\sigma }_{\mathit{pc}}+{\sigma }_{\mathit{ac}}\right)+\left({\sigma }_{\mathit{bc}}+{\sigma }_{\mathit{b},\mathit{HG}}\right)$$

(4)

where σ_pc is the axial normal stress caused by effective prestress; σ_ac is the axial normal stress caused by cable force; σ_bc is the bending normal stress caused by cable force; and σ_b,HG is the bending normal stress caused by hanger force (T_H) and self-weight (G). The bending normal stress caused by prestress is ignored because the prestressed tendons are symmetrically arranged in straight lines at the top and bottom flanges of the box cross-section of the girder.

Further study on the mechanical properties of the stiffening girder was conducted to analyze the primary factors that lead to the stress changes of the girder in the process of shrinkage and creep. Based on the finite element model, the stiffening girder was extracted and exerted with separate actions to analyze the stress changes caused by each action.

Table 3. The content of each finite element model analysis.

Model Number	Content of Analysis	Actions be Calculated
1	Change of σx	Npe + Tc + TH + G + 50 years’ effects of shrinkage and creep
2	Change of σac	Tc + TH + G + 50 years’ effects of shrinkage and creep
3	Change of σpc	Npe + TH + G + 50 years’ effects of shrinkage and creep
4	Change of σb,HG	TH + G + 50 years’ effects of shrinkage and creep
5	Change of σbc	Tc + 50 years’ effects of shrinkage and creep

shows the content of each finite element model analysis.

Figure 6 shows the changes of normal stress in the lower margin of the stiffening girder. A similar trend was found for changes of σ_pc and σ_ac in Figure 6a,b. σ_pc and σ_ac decreased by 0.4 MPa and 0.5 MPa, respectively, in 50 years. Figure 6c shows that the amplitude of σ_b, HG variation is the greatest, and the stress reduction in sections H4, H12, H15, and H19 is 0.3 MPa, 0.8 MPa, 3.1 MPa, and 5.9 MPa, respectively. The amplitude of σ_bc is less than 0.1 MPa, as shown in Figure 6d.

Combining Equation (4), σ_x of the stiffening girder can be drawn by the superposition of σ_pc and σ_ac, σ_b,HG, and σ_bc. Figure 7 shows the change of σ_x, as well as its composition in each five-year period.

As shown in Figure 7, the reduction in σ_pc and σ_ac is almost invariable, and each occupied about (5~6)% of the total stress reduction. The reduction in σ_bc is less and accounted for 3% of the total stress reduction. The reduction in σ_b,HG caused by the reduction in the hanger force accounted for 85~87%, which proved that the reduction in the hanger force is the dominant factor for the decrease in the stress in the lower margin of the stiffening girder.

In view of the deflection result in Table 3, the shrinkage and creep of concrete leads to deflection in the middle span and a decrease in the hanger force, which caused the increase in the positive bending moment in the middle span. Subsequently, the compression stress in the lower margin continues decreasing until the generation of tensile stress.

4. Control Method for the Reasonable Bridge State of a Stiffening Girder

The studies above indicate that shrinkage and creep have strong influence on the stiffening girder; it is therefore necessary to take the long-term effects of shrinkage and creep in the operation into consideration when determining the reasonable bridge state. The reduction in the hanger force has the greatest impact on the mechanical properties of stiffening girders during operation; thus, increasing the hanger force of the initial completed bridge state by an appropriate percentage can lead to good results. The increase in the hanger force (usually less than 5% of the devised hanger force) will alleviate the impacts of shrinkage and creep on the hanger force and ensure the reasonableness of the final completed bridge state in 50 years.

After increasing by 2.5~5% of the devised hanger force, the stress in sections H15 and H19 is shown in

Table 4. Stress in the stiffening girder under the case of increased hanger force.

State of Bridge	Position of Section	Increased by 2.5%		Increased by 3%		Increased by 4%		Increased by 5%
		H15	H19	H15	H19	H15	H19	H15	H19
Stress of initial state/MPa	Upper margin	−5.19	−4.75	−5.05	−4.54	−4.91	−4.32	−4.63	−3.88
	Lower margin	−6.81	−6.72	−7.07	−7.09	−7.33	−7.46	−7.85	−8.21
Stress of final state/MPa	Upper margin	−6.95	−7.76	−6.91	−7.68	−6.86	−7.6	−6.76	−7.45
	Lower margin	−1.96	−0.12	−2.04	−0.23	−2.12	−0.35	−2.28	−0.59

.

The increase in the hanger force makes the middle span of the stiffening girder bear a negative bending moment, which leads to an increase in compression stress in the lower margin and a decrease in stress in the upper margin. The law of stress change in the process of increasing hanger force is opposite to that of shrinkage and creep and can alleviate the adverse impact of shrinkage and creep. As shown in Table 3, increasing the hanger force by 2.5% can keep the stiffening girder in the total cross-section compressed state at 50 years. After increasing by 5%, the compression stress in the lower margin of the stiffening girder is −0.59 MPa in the mid-span after 50 years. The 5% increase shows better results, but the percentage should not be too large in consideration of the convenience of the construction and the overall safe stress state of the whole bridge.

By increase the hanger force of Hunan Road Bridge, the stress state of the stiffening girder of the initial and final completed bridge states in Figure 8 shows the effectiveness of the method in improving the mechanical behavior during bridge operation.

In practical engineering, the initial state of the completed bridge can be controlled by adjusting the system transformation scheme and taking the reasonable cable force in the final state of the completed bridge and the linear change of the stiffening beam as the control index. The reasonable completion state control process of stiffened beam considering the long-term shrinkage and creep effect of concrete is shown in Figure 9.

Based on this control method, the hanger force of Hunan Road Bridge was increased by 5% (165 kN) in the process of system transform. The elevation of the stiffening girder in the mid-span was raised by 4.3 cm. The simulation results show that the total cross-section of stiffening girder maintains the compression stress state under the effects of shrinkage and creep for 50 years, and the bridge shows better working conditions during long-term operation.

5. Conclusions

(1)

The long-term effects of shrinkage and creep have adverse impact on the stiffening girder, especially the mid-zone, which occupies nearly 50% of the length of the middle span. The compression stress in the lower margin decreases continuously, and the tensile stress eventually emerges in the mid-zone. It is therefore necessary to take the long-term effects of shrinkage and creep into consideration when determining the finished bridge state and to ensure the safety of the structure and the normal use in the long-term operation.

(2)

In view of the effects of shrinkage and creep on the stiffening girder, the process showed that the stress changes from fast to slow and tends to become stable gradually within 50 years. The changes in the first 15 years accounted for 90% of the total variation. Thus, the stable stress state of the 50th year can be set as the target state for the determination of the finished bridge state.

(3)

The reduction in hanger force is the primary factor that causes a decrease in the compression stress in the lower margin of the mid-zone of the stiffening girder. The stress changes caused by the reduction in the hanger forcer accounted for approximately 85% of the total stress variation. The loss of prestress and the decrease in the cable force caused approximately 5–6% of the total stress variation.

(4)

Based on the stress state of a stiffening girder that takes the 50 years of effects of shrinkage and creep into consideration, the hanger force was increased by less than 5% to relieve the adverse effects and improved the tress state in the long-term operation. The utilization in the Hunan Road self-anchored symmetry suspension bridge verified the simplicity and practicality of this new control method and confirms that it can be implemented to guide the design and construction of similar bridges.