4.7 Construction Errors
4.7.2 Error in elevation of a segment
In this chapter an error in the elevation of the girder is assumed. The error can be adjusted either by changing the cable forces with retention operations or, in case of smaller discrepancies from the target condition, by changing the camber in the remaining segments. Both methods shall be demonstrated in the following.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports 136
4.7.2.1 Adjustment by cable forces
If the error in the girder elevation is large, it is necessary to retension the cables, which will change both the internal forces of the structure and the girder level. As in the example given before, it is assumed that Cable 4 is erected with wrong initial stressing. Due to wrong tensioning of the cable, the girder elevation must be adjusted. As it is the same case for the tension forces, if a large error occurs during the construction, the adjustment of the girder elevation must proceed immediately before continuing the erection. An overall adjustment may be performed before connecting both cantilevers to close the bridge. However, to exemplify the general influence and to show how to calculate the additional tension forces, the girder elevation shall be also corrected in the last stage as it was assumed in the chapter before.
Figure 4-35 showed the difference in the girder level after Cable 4 has been installed. The final displacement in the last stage of the forward analysis is given in the vector δfi_0. The displacement, as it has been obtained from the original system without changing the initial tension in Cable 4, is given in δfi_A. In these vectors the vertical displacements at Node 4, 12, 16, 20 and 22 and the horizontal displacement at the top of the pylon at Node 106 are represented by the 1st to 6th value respectively. The location of the nodes is given in Figure 4-1. It is assumed that the vertical displacement δfi_A is considered in the camber construction of the girder. Therefore, the additional displacement ∆δfi_A must be adjusted by retensioning operations.
The values below are given in [mm].
δfi_0
The influence matrix for the displacement can be found by using the Unknown Load Function offered by MiDAS again. The displacement of the according nodes is restricted and the influence matrix is calculated. As it has also been the case in the previous chapter, this influence matrix is based on internal forces and must be factorized and rearranged for the purposed operations. The calculation can be done in a spreadsheet to find the required matrix from.
The influence matrix for the external forces follows to (in [mm])
It should be noted that the given influence matrix is not exactly the matrix for the system to be adjusted. Because the influence matrix is evaluated with the Unknown Load Function, the system which is employed has a different deformation and therefore a different angle between the girder and the cables. The change in the angle influences the calculated vertical girder elevation. However, this difference is small enough to be neglected.
The upper and lower restriction values must be increased again, otherwise it is not possible to find a solution. The following vectors state the lowest possible limits.
δfi_up
The utilisation of the optimisation method gives the required additional cable forces ∆Sfi_star and the new calculated displacement δfi.
∆Sfi_star calculated by MiDAS are the same as determined before. However, there is still a discrepancy between the target value δfi_A and the final calculation of ∆δer. The highest divergence is in the centre of the girder at node number 22 of -5 mm. But as mentioned earlier, with the specified adjustment requirements for the girder nodes and the top of the pylon, it has not been possible to fine-tune the girder elevation closer to the target values.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports
Because of the focus on the girder elevation in this calculation, the change in cable forces has been neglected. Table 4-20 gives the cable forces after the girder elevation has been adjusted in comparison to the tension forces with originally designed cable tuning.
Last step [tonf]
Table 4-20: Cable forces due to elevation adjustment
Not only are the cables forces influenced by the restressing operations, the overall internal forces are redistributed in fact, too. Figure 4-38 describes the final moment distribution after the adjustments.
Figure 4-38: Final moment distribution after elevation adjustment [tonfm]
The moment distribution indicates a high moment at the anchorage point of Cable 5. In the centre of the bridge, a high discrepancy between the actual vertical displacement and the target displacement has been assumed, which must to be adjusted. Due to this large difference, the restressing operation causes this high moment at the anchorage point of Cable 5.
Nevertheless, this and the previous example show that, if the correction procedure mainly focuses on the cable forces, there will be a gap in the girder elevation and vice versa, so that correcting the girder elevation results in a wrong cable force. It is the designers purpose to find an appropriate balance between both. Cable forces and girder elevations must be within the allowable range in the final state to ensure reliability and serviceability of the structure.
4.7.2.2 Adjustment by camber
In this chapter, it is assumed that an error in the girder elevation has occurred due to a wrong installed segment. This may happen in case of a prefabrication inaccuracy of the individual segment. It can also be the case that at site operation, for example during the welding, an additional angle is induced to the newly installed segment. For the example it is assumed that girder G4 is erected with an error producing an extra vertical displacement of -50 mm after the installation.
In order to model this condition, the model must be slightly modified. At the start of the 4th segment (at Node 12, see Figure 4-1) a further node is created very close to the existing one.
Then segment 4 is defined to start with Node 200 and both girder parts are connected by an elastic link, as it can be seen in the figure below.
Figure 4-39: Elastic link in order to model an error in the girder elevation
For the elastic link, the type Gen is chosen, which allows to define six stiffness values, - three directions and three rotations. High values for all degrees of freedom are defined to generate a rigid connection. In the construction stage analysis, the elastic link and Node 200 is activated at the same time of installing the previous segment (Segment 3). This is important because otherwise the tangential activation of the segments and the recalculation of the following cambers will result in errors. At the tip of the girder, the real vertical displacement is -110.57 mm after activating Segment 4. In the corresponding step of the original analysis without errors, the value is -110.76 mm. The small discrepancy is neglected so that it proves the reliability of the proposed method. In the next construction step, the elastic link can be replaced by another elastic link with a changed rotational stiffness, allowing a further displacement in order to
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports 140
achieve the error value as it has been measured on the construction site. The additional displacement for this example is -50.19 mm, which includes the small discrepancy as mentioned.
The rotational stiffness value is found as given in the table below.
Stiffness [tonf*m/rad] *104
Vertical Displacement [mm]
Target [mm]
∆ Stiffness [tonf*m/rad] *104
New Stiffness [tonf*m/rad] *104
50 -43.01 -8.347 41.653
41.653 -51.63 +1.162 42.815
42.815 -50.23
-50.19
-/- -/-
Table 4-21: Required rotational stiffness obtained from MiDAS
In the stage of activating the elastic link with the calculated rotational stiffness, the vertical real displacement is -160.79 mm. In the next construction stage, the active elastic link must be replaced again by a rigid connection. There is also the possibility to model the error displacement with only one additional step. However, it is easier to control the different changes by using different construction stages.
In this example, it has been assumed that the segment is installed with an error in the girder elevation which goes downward. Due to the temporary installation of an elastic link and the self-weight of the girder, the required error can be modelled. This procedure is not possible in case of an upward error. In this case, an elastic link can be used but additionally an external force, which produces the required deformation, must be applied. The load must be removed in the next step and at the same time, a fixed connection between both girder parts must be activated.
Since the internal forces do not change, the moment distribution and the cable forces of both systems are identical. At the final state, the vertical deformation for the original system (as already given in Figure 4-34) and the system including the erection error are given below.
Vertical displacement [mm] of the girder (Case II model) Original system
System including construction error
Figure 4-40: Vertical displacement original system and system including error in girder elevation
Due to the error in connecting girder G4 with the already installed girder G3 and because of the tangential erection of the following segments, the girder elevation changes in the subsequent construction stages and results in the final condition as seen in the deformed plot. The error can be compensated by changing the camber of the following segments, compared to the initial design with no additional errors. By modelling the error as described before, the General Camber function can be used to calculate the changes in the remaining segments. For the node at the tip of the segments, Table 4-22 gives the initially designed camber data and the required changes due to the error.
N 1 N 4 N 7 N 9 N 12 N 16 N 20 N 22
Orig. system 0.00 -1.00 6.03 14.52 32.38 52.85 64.98 69.56 Error system 0.00 -0.98 6.04 14.51 32.35 103.02 165.39 195.08
Table 4-22: Fabrication camber data [mm]
Node 16 defines the end of Girder 4, which has been assumed to be installed inaccurately and therefore the camber of this segment is neglected or already included in the structure. Only the two remaining segments, which are not installed at this time of construction, can be changed and can adjust the error so the final structure has no deformation.
Figure 4-41: Fabrication camber [mm]
The fabrication camber data given in Table 4-20 is transformed into a graphical form including the error for the system.
Chapter 4: Example of a Cable-Stayed Bridge including temporary supports 142