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The girder of prestressed concrete continuous box girder bridge strengthened by a stay cable system has a very complex deformation mechanism, and it is difficult to establish accurate numerical model for prediction. For these problems, a prediction method based on the combination of ant colony algorithm and residual combination correction model is proposed. In this method, the measuring points are considered as the cities in TSP ant colony algorithm, the information function model and heuristic function model are constructed, the pheromone update mechanism and ant search mechanism are established, and the deformation prediction of the main girder based on ant colony algorithm prediction model is achieved. On this basis, in order to fit the random change process of main girder deflection better, the periodic function generated by harmonic transform and sine function are introduced to modify the predicted results, which makes up for the low precision defect of single model. The results show that compared with finite element method, unequal interval gray model (1,1) (UIGM (1,1)), one-time residual correction UIGM (1,1), Markov chain residual correction UIGM (1,1), and ant colony algorithm model, this model can reflect the space-time effect and casual fluctuation feature of the development of girder deformation better and also has higher prediction accuracy and efficiency. The mean relative error of the predictive value is 3.39%, the posterior error ratio is 0.060, and the accuracy level reaches level 1. This model provides a new way for the girder deformation prediction of bridge strengthened by the stay cable system.

It has become a worldwide problem that midspan lag and box girder cracks are common in prestressed concrete (PSC) continuous box girder bridges [

At present, the deformation calculation methods of bridge structure mainly include theoretical analysis method, model test method, and numerical simulation method, intelligent prediction method. Among them, the theoretical analysis often adopts the idealized mechanical model, resulting in the relatively big difference between the theoretical calculation results and the measured values [

Unequal interval grey model (1, 1) (UIGM (1, 1)) is a grey dynamic prediction model consisting of a single-variable, first-order differential equation; it is a popular model of grey system theory and has no special requirements and limitations on experimental data, so it has a very wide application field [

Ant colony algorithm (ACA), a new heuristic bionic algorithm put forward by Dorigo M. et al., has the capacity of solving the problems of “complexity”, “nonlinearity” and “uncertainty”, “high cost” and obtains good results in long-term settlement prediction of tunnel [

According to the analyses above, this paper proposes a prediction method based on the combination of ant colony algorithm and residual combination correction model (ACAC model) to predict the deformation of main girder. First, ant colony algorithm is used to solve the problems of “complexity”, “nonlinearity” and “uncertainty”, “high cost”, excavate the inherent rules of sampling data, and realize the quantitative prediction of the change in the next phase. Secondly, the periodic sequence function and sine function generated by harmonic variation are used to fit the residual sequence, which is used to modify ant colony algorithm model prediction value and compensate the deflect of low precision of the single prediction model, so that the model could achieve a higher prediction accuracy. The effect of this model was verified by the measured data of the main girder in the tensioning phase. The example analysis shows that the prediction accuracy of this model is better than that of UIGM (1, 1) model and ant colony algorithm model; this model has a certain theoretical and engineering application value.

Ant colony algorithm [

Combining the Traveling Salesman Problem (TSP), this paper introduces ant colony algorithm; its mathematical model can be described as follows: to set the number of cities in the plane to be

Suppose

Taboo list tabu_{k} (_{k} is constantly updated with the search process of the ant. During the traversal, the ant calculates state transition probability based on the number of pheromones and the elicitation information in each route [

In the formula, _{k} represents the collection of candidate cities that ant

The heuristic function

In the formula,

When the ants are traveling in the city, the pheromones are constantly updated according to formulae (

In the formula,

Three models were developed by Colorni A. et al. according to the pheromone update method, namely, Ant-Cycle model, Ant-Quantity model, and Ant-Density model [

Ant-Cycle model:

Ant-Quantity model:

Ant-Density model:_{k} is the path length of the ant

The Ant-Cycle model uses the overall information; that is, the ant updates the pheromone on each path after a complete tour. The latter two models rely on local information; that is, the ant updates the pheromone in every step. The previous research has shown that the Ant-Cycle model outperforms the other two in dealing with the TSP [_{k} of ant

According to the deformation curve of the main girder in three continuous tensioning phases, the deformation difference between the last two phases of a measuring point is closely related to (1) the deformation difference between the first two phases of this measuring point and (2) the result of the deformation difference between the first two phases of another measuring point minus that of yet another measuring point. This relationship is consistent with the positive feedback mechanism of the ACA; that is, the present behaviour can strengthen the future behaviour. Hence, the deformation difference between the first two phases of the main girder was analogized as the present behaviour and taken as the input parameters of the ACA prediction model. On this basis, it is possible to predict the deformation difference between the last two phases in an accurate manner.

Establishing reasonable heuristic function and information function is the key of constructing the deformation prediction model of the reinforced girder. Among them, the information function reflects the influence degree of the whole ant colony’s cumulative experience on ants’ path selection, and the inspiration function reflects the degree of influence of the ants’ individual experience on selecting path. In order to apply the basic idea of ant colony algorithm into the deformation of the girder strengthened by a stay cable system, it is necessary to establish a similar colony path selection mechanism according to the actual deformation of the early tension phase of the main girder. In this paper, the monitoring points of the bridge are considered as the cities where the ants travel in TSP; in the course of selecting the path, the ant tends to move towards the measuring point of the larger deformation. As a result, the cumulative deformation of a site is regarded as an information function which can reflect the whole ant colony’s cumulative experience. However, the difference of cumulative deformation of a certain measurement point in each period can be regarded as an inspiration function which reflects the experience of an individual ant. After calculating the transition probability based on the information function and the heuristic function of the path, ants move to the next point and release a certain concentration of pheromones on the path that they have passed in order to increase the cumulative experience of the whole ant colony. Thus, the positive feedback and auto-catalytic mechanism of path selection of ant colony algorithm is accomplished.

Based on this, the initial information function

In the formula,

Unlike the TSP, the ant movement of the ACA depends on the deformation difference of measuring point between tensioning phases rather than the path length. The pheromone update method is similar to that in Ant-Quantity model; i.e., the ants update the pheromone in each step. Thus, the concentration of pheromone

It can be seen from formula (

Obviously, the value of

By using the parallel distributed search capability of ant colony algorithm, several ants are placed in the girder deformation area. These ants are ordered to search and update pheromones based on the aforementioned path selection mechanism and pheromones updating mechanism and also take self-organizing evolution of the algorithm.

At the end of the ant search, the number of times the ants arrive at each measuring point between tensioning phases

The total number of times the ants arrive at all measuring points is denoted as

Then, obtain the predicted cumulative deformation in tensioning phase

The performance of ant colony algorithm prediction model is affected by parameters

Determine the ants’ total number

Adjust the parameters with a larger value range in turn; they are

Adjust the parameter

Ant colony algorithm prediction model can realize the prediction of the main girder deformation in the third equal interval tensioning phase with only two sets of deformation measured values at adjacent equal interval phases as input data. Because there might be large difference between the predicted deformation values of ant colony algorithm and the measured values, it is necessary to take a step further to use the residual correction model to modify the model and make up for the defect of low accuracy and poor adaptability of single prediction model.

Generally, the residual sequence of the main girder deformation is usually alternatively positive and negative and has periodicity which is not so regular. This phenomenon means the original data column has some random components and periodic components. In order to further improve the prediction accuracy of the model, generated by harmonic variation, the periodic sequence function and sine function are used to modify the model, and the former can reflect most of the periodic components of the residual sequence waveform, while the latter can reflect the remaining small fraction of random fluctuations in the waveform and be supplement to the former [

Build residual sequence in

So, residual sequence of

Fit

Fit

Introduce the weight coefficient

Based on the nearby principle of data, the most recent data can reflect the nature of main girder deformation.

Initialize the parameters

Determine the initial position of ants. First of all, according to the number of measuring points

Calculate the initial information function

After the termination of search, the predicted deformation values of measuring points in next phase can be calculated by formulae (

According to a certain step length, parameter

Select

Add up the ant colony algorithm model prediction value

The flowchart of deformation forecasting process of SCS-enhanced main girder in tensioning phase by ACA correction model is shown in Figure

Deformation prediction flowchart of SCS-enhanced main girder in tensioning phase by ACA correction model.

The main bridge of Dongming Huanghe River Highway Bridge belongs to PSC continuous rigid frame-continuous beam composite structure system; it has nine holes and one link, and the cross-diameter combination is

Strengthening of Dongming Huanghe River Highway Bridge by application of a stay cable system.

In order to grasp the deformation of the main bridge during the tensioning phases, 46 vertical displacement measuring points were set on the upstream and downstream of the whole bridge. This paper selects the upstream monitoring data to study, as shown in Table

Vertical deformation of the measuring points under different tensioning phases.

Measured points | Distance from 57#pier center | Vertical deformation | ||||
---|---|---|---|---|---|---|

630kN | 1,155kN | 1,365kN | 1,575kN | 1,785kN | ||

C1 | 37.5 | -9.13 | -8.48 | -7.73 | -7.45 | -6.31 |

C2 | 105.0 | 17.77 | 21.76 | 22.82 | 24.39 | 27.79 |

C3 | 135.0 | 35.82 | 44.65 | 47.25 | 50.92 | 52.25 |

C4 | 165.0 | 17.77 | 20.21 | 22.66 | 25.39 | 27.12 |

C5 | 225.0 | 4.58 | 7.53 | 9.35 | 9.86 | 10.97 |

C6 | 255.0 | 20.61 | 33.27 | 35.11 | 37.87 | 37.87 |

C7 | 285.0 | 7.38 | 12.18 | 13.01 | 14.95 | 16.84 |

C8 | 345.0 | 7.55 | 7.96 | 8.23 | 8.94 | 10.48 |

C9 | 375.0 | 17.73 | 23.69 | 26.57 | 30.84 | 32.51 |

C10 | 405.0 | 9.10 | 11.75 | 13.58 | 15.21 | 16.99 |

C11 | 465.0 | 7.56 | 9.27 | 9.76 | 11.54 | 12.18 |

C12 | 495.0 | 16.11 | 21.31 | 23.82 | 27.81 | 29.44 |

C13 | 525.0 | 5.54 | 7.52 | 8.15 | 10.15 | 11.66 |

C14 | 585.0 | 7.61 | 9.82 | 9.99 | 10.35 | 10.38 |

C15 | 615.0 | 18.98 | 24.76 | 27.17 | 30.49 | 32.11 |

C16 | 645.0 | 7.03 | 8.16 | 8.53 | 9.06 | 9.25 |

C17 | 705.0 | 8.04 | 12.64 | 13.33 | 14.02 | 16.45 |

C18 | 735.0 | 16.48 | 25.34 | 28.77 | 31.48 | 33.20 |

C19 | 765.0 | 4.18 | 8.05 | 9.68 | 10.03 | 11.45 |

C20 | 825.0 | 11.92 | 15.06 | 15.37 | 16.73 | 17.88 |

C21 | 855.0 | 25.30 | 33.27 | 37.15 | 42.78 | 45.18 |

C22 | 885.0 | 11.05 | 16.29 | 17.88 | 19.45 | 21.57 |

C23 | 952.5 | 6.09 | 7.83 | 8.06 | 8.24 | 8.26 |

This paper selects the measuring deformation data of 1,575kN and 1,365kN tensioning phases and predict the deformation in 1,785kN tensioning phase

Then, determine the number of ants

When all the ants reach the detection point C21, the search terminates, and the number of times the ants arrive at each measuring point is as follows:

According to formula (

According to formula (

Based on the prediction model of ant colony algorithm, the residual sequence is as follows:

There is a certain deviation between the predicted and measured deformation values in the 1,785kN tensioning phase. This is attributable to the following two reasons. First, the ants are not sensitive to the measuring points with small deformation in the early phase; thus, some measuring points are not visited by the ants, leading to a large residual value. Second, it can be seen from the calculation results of the residual sequence of 1,785kN tensioning phase that this residual sequence is usually alternatively positive and negative and has periodicity which is not so regular. This phenomenon means the original data column has some random components and periodic components.

In order to further improve the prediction accuracy of the ant colony algorithm prediction model, this paper takes fitting calculation on residual sequence of 1,575kN tensioning phase by residual combination correction model and takes it as the correction sequence of the predicted value of the 1,785kN tensioning phase.

Firstly, based on the ant colony algorithm prediction model, the deformation prediction value

Then, the residual sequence is as follows:

Secondly, the residual sequence is modified by sine function, and the parameter ^{2}, the calculation result is best. The first 300 iteration results are shown in Figure

Relation graph between parameter

Then, the periodic sequence function and sine function generated by harmonic variation are used to modify the residual sequence in combination. Set ^{2}, the calculation result is best; the first 300 iteration results are shown in Figure

Relation graph between parameters

So, the correction sequence of the 1,785kN tensioning phase is as follows:

Finally, the deformation under the ant colony algorithm correction model is obtained by formula (

This paper regards the cable force data in tensioning phase as the original time sequences, regards the vertical deformation data obtained from the bridge floor survey points as the original date sequences, and builds the prediction model of main girder deformation strengthened by a stay cable system based on UIGM (1, 1) [

In addition, the modified model provided in [

Under various factors, the main girder deformation and its trend cannot be determined completely, which is consistent with the highly random nature of Markov residual prediction. Through Markov residual prediction, it is possible to improve the random features of the sequence and optimize the predicted results. Therefore, the UIGM (1, 1) predicted value is modified by the Markov chain in [

Here, measuring point C3 is cited as an example to describe the prediction process. According to the predicted results of UIGM (1, 1), the residual sequence of the first four tensioning phases was identified as

The residual sequence was transformed into a nonnegative one by adding a translation quantity H.

The new residual was predicted again.

The predicted value was subtracted by the translation quantity H.

After substituting (630, 1,155, 1,365 and 1,575) and (5.00, 4.60, 5.46 and 8.11) into the UIGM (1, 1), the residual value was predicted as 6.79mm and 1.79mm after the translation transformation.

Thus, the modified predicted deformation value is as follows:

Generally, the state intervals are classified based on the mean and standard deviations of the data sequence, and the mean

The residual sequence was divided into three state intervals

State of residual sequence.

Tensioning force/kN | Residual/mm | State |
---|---|---|

630 | 0.00 | |

1,155 | -0.40 | |

1,365 | 0.46 | |

1,575 | 3.11 | |

The residual value of the 1,785kN tensioning phase was predicted by the state transitions of 630kN~1,575kN tensioning phases, with a total of 3 steps of transitions. The state transition matrix can be expressed as

Then, the transition probabilities are listed in Table

State transition probabilities.

Transition step | Tensioning force/kN | State | Transition probability | ||
---|---|---|---|---|---|

| | | |||

1 | 1,575 | | 0 | 0 | 0 |

2 | 1,365 | | 0 | 1 | 0 |

3 | 1,155 | | 0 | 0 | 0 |

The state centres were calculated as

For other measuring points, the deformation in the 1,785kN tensioning phase can be obtained similarly with the OC model and MC model.

The previous studies have attributed the deformation of PSC box girder bridges to the following factors: concrete shrinkage, creep, prestress loss, and the mismatch between actual and theoretical concrete amounts [

The deformation features of the SCS-enhanced main girder in the tensioning phase were discussed using the finite-element software. A finite-element model was established considering the following factors: (1) the section construction in 1991; (2) the effective prestress of tendon determined by actual measurement and theoretical calculation; (3) time-dependent effect of concrete materials; (4) shear stiffness reduction of the web; (5) the addition of external tendons and increase of web thickness in 2003; (6) the mismatch between actual and theoretical concrete amounts; (7) tensioning sequence of stay cables.

In the finite-element model, the main girder was discretized into 347 girder elements, the stay cable into 64 truss elements, and the bridge pier into 312 girder elements. The shrinkage and creep of concrete were simulated according to the

Finite element model of main girder strengthening by a SCS.

The deformation increment caused by the diagonal cracks of the web was simulated by the shear stiffness degradation model [

(1) The main bridge was divided into several units. Then, the grade

Diagonal crack grading standard.

Diagonal crack scale | Criteria | |
---|---|---|

Qualitative description | Quantitative description | |

1 | There is no diagonal crack. | - - |

2 | There are a few slight diagonal cracks, whose width falls within the allowable range. | The crack length is less than 1/3 of the sectional size. |

3 | There are many diagonal cracks, whose width falls within the allowable range. | The crack length is between 1/3 and 1/2 of the sectional size. |

4 | There are numerous diagonal cracks, whose width exceeds the allowable range. | The crack length is greater than 1/2 of the section size, and the mean crack interval is less than 30cm. |

5 | There are numerous through diagonal cracks, whose width far exceeds the allowable range. | The crack width is greater than 1.0mm and the mean crack interval is less than 20 cm. |

(2) The limit

(3)The SSDFs

(4) The modified Poisson’s ratios were determined for all crack scales:

(5) The shear stiffness degradation was simulated by modifying Poisson’s ratio of each unit in the finite-element model, laying the basis for deformation analysis of the main girder.

The calculated SSDF and modified Poisson’s ratio of our example are listed in Table

Parameter selection for different grade scales of web diagonal cracks.

Diagonal crack scale | Shear stiffness degradation factor | Modified Poisson’s ratio |
---|---|---|

1 | 1.00 | 0.20 |

2 | 0.79 | 0.52 |

3 | 0.58 | 1.07 |

4 | 0.36 | 2.33 |

5 | 0.15 | 7.00 |

According to the FEM results, the deformation of the measuring points in the 1,785kN tensioning phase amounts to

The FEM predicted value was contrasted with the value measured in the 1,785kN tensioning phase, revealing that it deviated from the measured value by 23.83%. The deviation comes from two possible reasons. First, the FEM failed to simulate the deformation of measuring point C1 in the 630kN tensioning phase, leading to a 252.19% error between the predicted and measured values in the 1,785kN tensioning phase. Second, the established model did not consider the deformation increment that resulted from the long-term structural damage under vehicle load, bar corrosion, prestressed tendon corrosion, and temperature effect. The predicted values of all measuring points other than C1 were smaller than the measured values. The variation trends of the measuring points were consistent and the mean relative error was 13.45%.

The deformation values and relative error obtained by six models are as shown in Table

Comparison of predicted and measured values.

Measured points | Measured values/mm | FEM | UIGM (1,1) | OC - UIGM (1,1) | MC - UIGM (1,1) | ACA model | ACAC model | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Predicted | Relative | Predicted | Relative | Predicted | Relative | Predicted | Relative | Predicted | Relative | Predicted | Relative | ||

values/mm | error/% | values/mm | error/% | values/mm | error/% | values/mm | error/% | values/mm | error/% | values/mm | error/% | ||

C1 | -6.31 | 9.60 | 252.19 | -7.66 | -21.32 | -7.33 | -16.16 | -7.26 | -15.06 | -7.45 | -18.07 | -6.40 | -1.43 |

C2 | 27.79 | 24.92 | 10.33 | 23.50 | 15.44 | 24.20 | 12.92 | 23.83 | 14.25 | 26.78 | 3.62 | 27.76 | 0.12 |

C3 | 52.25 | 46.90 | 10.25 | 48.85 | 6.51 | 50.64 | 3.08 | 49.64 | 5.00 | 53.49 | -2.37 | 51.82 | 0.83 |

C4 | 27.12 | 23.40 | 13.72 | 21.91 | 19.20 | 24.75 | 8.74 | 25.18 | 7.15 | 27.96 | -3.08 | 27.27 | -0.54 |

C5 | 10.97 | 10.56 | 3.70 | 9.64 | 12.12 | 10.42 | 5.01 | 10.52 | 4.10 | 9.86 | 10.12 | 11.72 | -6.85 |

C6 | 37.87 | 31.92 | 15.70 | 36.31 | 4.11 | 37.58 | 0.77 | 36.89 | 2.59 | 37.68 | 1.17 | 37.86 | 0.69 |

C7 | 16.84 | 13.30 | 21.03 | 13.88 | 17.58 | 14.57 | 13.48 | 14.22 | 15.56 | 17.52 | -4.01 | 18.29 | -8.60 |

C8 | 10.48 | 8.83 | 15.73 | 8.54 | 18.52 | 8.78 | 16.22 | 8.66 | 17.37 | 12.87 | -22.85 | 10.40 | 0.75 |

C9 | 32.51 | 27.27 | 16.12 | 28.55 | 12.18 | 30.62 | 5.81 | 29.44 | 9.44 | 33.41 | -2.76 | 30.71 | 5.55 |

C10 | 16.99 | 15.39 | 9.43 | 14.36 | 15.50 | 15.38 | 9.48 | 15.66 | 7.83 | 17.60 | -3.62 | 16.24 | 4.42 |

C11 | 12.18 | 11.96 | 1.79 | 10.56 | 13.32 | 11.09 | 8.95 | 11.59 | 4.84 | 13.42 | -10.19 | 12.50 | -2.60 |

C12 | 29.44 | 26.94 | 8.49 | 25.67 | 12.81 | 27.51 | 6.56 | 26.83 | 8.87 | 30.38 | -3.18 | 28.74 | 2.39 |

C13 | 11.66 | 9.48 | 18.67 | 9.07 | 22.23 | 9.69 | 16.90 | 10.23 | 12.26 | 12.72 | -9.06 | 12.55 | -7.61 |

C14 | 10.38 | 8.14 | 21.54 | 10.14 | 2.28 | 10.27 | 1.06 | 10.21 | 1.64 | 10.35 | 0.29 | 9.99 | 3.75 |

C15 | 32.11 | 26.62 | 17.10 | 28.67 | 10.70 | 30.32 | 5.57 | 29.39 | 8.47 | 33.06 | -2.95 | 30.77 | 4.18 |

C16 | 9.25 | 7.07 | 23.52 | 8.76 | 5.32 | 8.99 | 2.81 | 8.87 | 4.11 | 9.06 | 2.05 | 9.43 | -1.91 |

C17 | 16.45 | 14.96 | 9.07 | 13.62 | 17.17 | 14.00 | 14.89 | 13.80 | 16.11 | 16.07 | 2.29 | 15.06 | 8.43 |

C18 | 33.20 | 28.97 | 12.73 | 30.03 | 9.55 | 31.99 | 3.64 | 30.83 | 7.14 | 34.05 | -2.55 | 33.64 | -1.33 |

C19 | 11.45 | 9.23 | 19.35 | 9.87 | 13.80 | 10.54 | 7.95 | 10.64 | 7.07 | 10.03 | 12.40 | 11.94 | -4.25 |

C20 | 17.88 | 14.11 | 21.09 | 15.96 | 10.72 | 16.32 | 8.72 | 16.74 | 6.38 | 18.95 | -6.01 | 17.65 | 1.26 |

C21 | 45.18 | 37.89 | 16.13 | 39.75 | 12.02 | 42.59 | 5.73 | 40.94 | 9.38 | 47.91 | -6.05 | 46.44 | -2.80 |

C22 | 21.57 | 21.28 | 1.37 | 18.58 | 13.85 | 19.49 | 9.64 | 19.79 | 8.25 | 19.96 | 7.45 | 20.51 | 4.92 |

C23 | 8.26 | 7.51 | 9.07 | 8.13 | 1.53 | 8.25 | 0.12 | 8.29 | -0.36 | 8.24 | 0.24 | 7.97 | 3.53 |

In order to reflect the data in Table

Predicted results of different models.

Predicted relative error of three different models.

The prediction models were contrasted and evaluated from three aspects: the measured datasets for modelling (MDM), the time consumption, and the mean absolute error (MAE), as shown in Table

Comparison of evaluation results under different prediction models.

Prediction model | MDM | Time consuming/s | MAE/% |
---|---|---|---|

FEM | 0 | 1196.87 | 23.83 |

UIGM(1,1) | 4 | 2.07 | 12.51 |

OC - UIGM (1,1) | 4 | 4.18 | 8.01 |

MC - UIGM(1,1) | 4 | 2.76 | 8.40 |

ACA model | 2 | 0.05 | 5.90 |

ACAC model | 3 | 0.13 | 3.39 |

It can be concluded from Table

The proposed model enjoys the following advantages over the five contrastive prediction models:

(1) Compared with UIGM (1, 1), the model can disclose the deformation trends of multiple correlated measuring points.

(2) Compared with the grey modelling method, the model requires a limited amount of modelling data.

(3) Compared with the four models, the model consumes a very short computing time.

(4) Compared with the five models, the model achieves a very low mean absolute error (3.39%).

Of course, the model still has some limitations:

(1) Unlike the FEM, the model must be modelled based on the measured data.

(2) The model needs one more dataset than the ACA.

(3) Unlike UIGM (1, 1), the model only works when the measured data are continuous and of equal interval.

In order to further determine whether the correction model is reliable for the deformation prediction of the main girder, the model accuracy test is carried out by using statistical posterior error

To sum up, the proposed model can achieve fast computation and high prediction accuracy using a few measured data through easy programming and simple operation. The case study reveals that the model is an excellent tool for predicting SCS-enhanced main girder deformation in the tensioning phase, shedding new light on main girder deformation control and prediction.

In the same way, the deformation values of the next tensioning phase (1,995kN) are extrapolated. Table

Comparison of predicted values in 1,995kN tensioning phase.

Measured points | Predicted values/mm | |||||
---|---|---|---|---|---|---|

FEM | UIGM(1,1) | OC - UIGM (1,1) | MC - UIGM(1,1) | ACA model | ACAC model | |

C1 | 11.37 | -7.25 | -6.86 | -6.79 | -4.61 | -5.03 |

C2 | 28.31 | 24.91 | 25.73 | 25.89 | 31.19 | 30.46 |

C3 | 52.93 | 50.30 | 52.47 | 52.63 | 53.95 | 55.27 |

C4 | 26.27 | 25.29 | 28.74 | 29.03 | 28.82 | 30.31 |

C5 | 13.12 | 10.25 | 11.16 | 11.26 | 12.67 | 11.61 |

C6 | 37.09 | 37.11 | 38.62 | 38.83 | 37.98 | 40.99 |

C7 | 15.97 | 14.96 | 15.77 | 15.68 | 18.54 | 20.02 |

C8 | 11.20 | 9.16 | 9.43 | 9.12 | 10.48 | 13.19 |

C9 | 32.10 | 30.34 | 32.81 | 32.98 | 32.62 | 33.95 |

C10 | 17.90 | 15.44 | 16.64 | 16.93 | 18.69 | 19.81 |

C11 | 14.29 | 11.22 | 11.83 | 12.42 | 13.88 | 15.22 |

C12 | 31.58 | 27.35 | 29.54 | 29.04 | 29.55 | 30.94 |

C13 | 11.82 | 10.07 | 10.79 | 11.42 | 12.45 | 13.73 |

C14 | 10.55 | 10.25 | 10.40 | 10.38 | 10.38 | 10.67 |

C15 | 31.24 | 30.14 | 32.10 | 32.25 | 32.22 | 33.43 |

C16 | 9.29 | 8.96 | 9.23 | 9.30 | 12.20 | 11.85 |

C17 | 17.60 | 14.52 | 14.97 | 15.02 | 18.15 | 18.46 |

C18 | 34.01 | 31.47 | 33.85 | 33.61 | 34.11 | 35.20 |

C19 | 11.67 | 10.50 | 11.29 | 11.38 | 13.15 | 11.78 |

C20 | 16.82 | 16.63 | 17.06 | 17.54 | 19.58 | 21.58 |

C21 | 43.67 | 42.16 | 45.67 | 45.67 | 46.88 | 49.95 |

C22 | 24.52 | 19.70 | 20.78 | 21.09 | 23.27 | 21.78 |

C23 | 9.39 | 8.20 | 8.33 | 8.38 | 8.26 | 8.78 |

It is difficult to accurately predict the girder deformation of PSC continuous box girder bridges enhanced by the SCS, owing to the complex deformation mechanism. The existing prediction models, lacking the modelling based on measured data, face low accuracy and poor adaptability in application. To overcome these defects, this paper combines the ant colony algorithm and the combined residual correction model into a new prediction model, which not only tackles the complexity, nonlinearity, uncertainty, and high cost, but also mines out the inherent deformation rules of main girder under the tensioning force. The new model was compared with finite element model, UIGM (1, 1), one-time residual correction UIGM (1,1), Markov chain residual correction UIGM (1,1), and ant colony algorithm model. The main conclusions are as follows.

(1) Compared with the unequal interval GM (1,1), one-time residual correction UIGM (1,1), and Markov chain residual correction UIGM (1,1), ant colony algorithm prediction model takes into account the time-space effect of deformation development of multiple points, and it can better dig out the inherent rules of the original data and make the prediction results more accurate, and its mean absolute error is 5.90%.

(2) Using residual combination correction model to modify the predicted result of the ant colony algorithm can well simulate the trend components and random fluctuation compositions of the main girder deformation, and it has high prediction accuracy and the mean absolute error of perdition values is 3.39%, the posterior error ratio is 0.060, and the accuracy level reaches level 1. The prediction precision is improved greatly; the paper’s model has a certain theoretical and engineering application value.

(3) The model needs less measured data, the principle of the model is simple, and it is easy for programming, but the theory of optimal parameter selection needs to be further improved in the follow-up work so as to improve its forecasting efficiency and expand the range of adaptation. Unlike the finite element model, the model must be modelled based on three continuous and of equal interval measured data.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of the paper.

This work presented herein has been supported by the National Natural Science Foundation of China (No. 11372165) and Project of Science and Technology of Western Region Transportation Construction, Ministry of Transport (No. 2011318223940).