Sensitivity analysis of construction control parameters for long-span continuous rigid‑frame bridges with low piers

2.1. Finite element modeling and parameter selection

A finite element model is established using Midas Civil. The entire bridge structure is simulated using beam elements, with a total of 676 elements and 709 nodes. Rigid connections are adopted between the main piers and the main girder, while general connections are used between the transition piers and the main girder. In addition, to account for the flexibility effect of the group pile foundations, a foundation model considering pile-soil interaction is established based on the Winkler model [8]. The finite element model is shown in Fig. 3. The main girder is made of C55 concrete with an elastic modulus of $E=$3.45×10⁴ MPa; the piers are made of C50 concrete with $E=$3.55×10⁴ MPa; and the prestressing tendons are made of ${\phi }_s=$ 15.2 steel strands, with a tensioning control stress of ${\sigma }_{con}=$1395 MPa and an elastic modulus of $E=$1.95×10⁵ MPa.

Fig. 3Full-bridge finite element model

2.2. Parameter sensitivity analysis method

During the cantilever construction of long-span continuous rigid-frame bridges, factors such as structural unit weight, elastic modulus, and prestressing tensioning all increase the complexity of construction control. In the design phase, structural analyses are often performed based on assumed parameters to predict the stress and deflection at each construction stage, thereby ensuring that the completed bridge state satisfies the design requirements. However, the actual parameter values often deviate from the design assumptions, causing the structural responses to deviate from the expected values, which may lead to errors in the internal forces and deflections at the completion stage. Therefore, the core of construction control lies in adjusting the parameters to make the analytical model approximate the actual structural conditions. Since different parameters exert varying degrees of influence on the structural responses, it is necessary to identify the dominant and secondary parameters through sensitivity analysis. Among these, the dominant parameters should be strictly controlled during construction, while the secondary parameters should also be given due attention when higher accuracy is required.

In bridge construction monitoring, parameter sensitivity analysis aims to quantify the influence of each parameter on the structural deflection and stress responses. The single-variable method is adopted to calculate the variations in structural responses, and a sensitivity coefficient is introduced for quantitative assessment. Its expression is given as follows:

where $df$ denotes the response under the design state, $\mathrm{\Delta }df$ is the variation in the response caused by a single-parameter change, and $r$ is the relative variation amplitude of the parameter. A larger value of $S$ indicates higher parameter sensitivity.

3.1. Elastic modulus of concrete

The structural stiffness of long-span continuous rigid-frame bridges is primarily governed by the elastic modulus of the material and the sectional properties. In this section, the elastic modulus of concrete is taken as 0.9E, 1.0E, and 1.1E, respectively, to analyze the variations in the main girder deflection and key section stresses under the maximum cantilever stage of a single T-shaped structure. The results are shown in Fig. 4 and Table 1, and the deflection sensitivity coefficients are presented in Fig. 5.

Fig. 4Main girder deflection at the maximum cantilever stage

Fig. 5Deflection sensitivity coefficients at maximum cantilever stage

As shown in Fig. 4 and 5 and Table 1, when the elastic modulus varies by 10 %, the deflection of the main girder at the cantilever stage decreases with increasing elastic modulus, with a maximum change of 6.55 mm, accounting for 9.3 % of the deflection at that location. The maximum stress change at the upper flange of the root section is 0.10  MPa, with a corresponding sensitivity coefficient of 0.07, while the stress at the lower flange remains virtually unchanged. The deflection sensitivity coefficients of each segment are more pronounced when the elastic modulus decreases, with Segments 10-13 exhibiting the highest sensitivity. Overall, the elastic modulus has a significant influence on the deflection of the main girder, but a relatively minor effect on the stress, and the deflection variation tends to accumulate with increasing cantilever length.

3.2. Unit weight of concrete

In long-span structures, the self-weight accounts for over 70 % of the total load, so variations in unit weight have a significant effect on the structural forces and deflections. In this section, the unit weight is taken as 0.9$\gamma$, 1.0$\gamma$, and 1.1$\gamma$, respectively, to calculate the main girder deflection and key section stresses under the maximum cantilever stage of a single T-shaped structure. The results are shown in Fig. 6 and Table 2, and the deflection sensitivity coefficients are presented in Fig. 7.

Fig. 6Main girder deflection at the maximum cantilever stage

Fig. 7Deflection sensitivity coefficients at maximum cantilever stage

As shown in Fig. 6 and 7 and Table 2, when the unit weight of concrete varies by 10 %, the deflection of the main girder at the cantilever stage decreases with increasing unit weight, with a maximum change of 20.27  mm, accounting for 29.38 % of the deflection at that location. The maximum stress change at the upper flange of the root section is 1.50  MPa, with a corresponding sensitivity coefficient of 1.03, while the stress change at the lower flange is 1.56  MPa, with a sensitivity coefficient of 1.50. The deflection sensitivity of each segment to both increases and decreases in unit weight is generally comparable, with the deflection at the maximum cantilever segment being the most sensitive, considerably higher than that of the other segments. In summary, variations in unit weight have a significant influence on both the deflection and stress of the main girder, and thus should be strictly controlled during construction.

3.3. Prestressing tensioning

Prestressing serves as a key component of long-span continuous rigid-frame bridges, primarily used to resist tensile stresses in the girder. However, during actual construction, issues such as insufficient tensioning or over-tensioning of the prestressing tendons may occur, leading to deviations between the actual in-girder prestress and the design values.

In this section, the in-girder prestress is taken as 0.9${\sigma }_{con}$, 1.0${\sigma }_{con}$, and 1.1${\sigma }_{con}$, respectively, to calculate the main girder deflection and key section stress variations under the maximum cantilever stage of a single T-shaped structure. The results are shown in Fig. 8-9 and Table 3 (deflection sensitivity coefficients).

Fig. 8Main girder deflection at the maximum cantilever stage

Fig. 9Deflection sensitivity coefficients at maximum cantilever stage

As shown in Fig.8 and 9 and Table 3, when the prestressing level varies by 10 %, the deflection of the main girder at the cantilever stage decreases with increasing prestressing, with a maximum change of 11.57  mm, accounting for 18.75 % of the deflection at that location. The maximum stress change at the upper flange of the root section is 2.80  MPa, with a corresponding sensitivity coefficient of 1.93, while the stress change at the lower flange is 0.53  MPa, with a sensitivity coefficient of 0.51. The deflection sensitivity of each segment to both increases and decreases in prestressing is generally consistent, with Segments 8~12 exhibiting the highest sensitivity, considerably higher than that of the other segments. In summary, variations in prestressing have a significant influence on both the deflection and stress of the main girder, and thus insufficient tensioning or over-tensioning should be avoided during construction.

3.4. Limitations of the univariate method

This study employs a univariate sensitivity analysis, varying each parameter independently by ±10 % while keeping the others at their design values, to reveal the dominant sensitivity ranking of elastic modulus, concrete unit weight, and prestress tensioning. However, this method cannot capture coupling and interaction effects under simultaneous multi-parameter deviations; for example, the deflection and stress responses induced by concurrent fluctuations in concrete unit weight and prestress tensioning are not simply additive. Future studies should adopt a multivariate analysis framework to quantitatively characterize parameter interactions. Nevertheless, univariate analysis can still provide a basis for identifying controlling parameters that require priority monitoring.