Cantilever concreting technology is one of the modern methods of constructing concrete long-span bridges. Characteristic feature of those bridges is the long-term span deflection resulting from the rheological processes in the concrete and in the pre-tensioning steel. It can also be caused by the material deterioration, e.g. concrete cracking, as well as the changes in the bridge structure, such as the support settlements. The aggregate result of bridge exploitation are the changes in its grade line, considered in this paper as the bridge span deflection line. The aim of the paper is the assessment of the internal forces on the basis of the bridge span deformation. Furthermore, an algorithm for the correction of the deflection function determined on the basis of surveying measurements (low precision measurements) is proposed. It is characterized by a significant improvement of the computational results, and it hardly “smoothens” the primary measurement results. The algorithm can be used to analyse the selected part of the bridge structure, e.g. the longest span. The paper proposes a universal coefficient of cantilever deflection, which is calculated on the basis of the cantilever joint moment when the final static scheme of the bridge is created. It can be used for the comparative analyses of various bridges. The value of the coefficient is dependent on the geometry of the cantilever box cross-section only.

Cantilever concreting technology in bridges was first used in 1951 by U. Finsterwalder during the building of a bridge over the Lahn Bulduinstein River. In these times, bridges constructed using this technology did not usually exceed half of the designed 100-year service life. Recently, several thousand bridges of this type have been made in the world e.g. [

Cantilever concreting (or assembling) technology is one of the modern methods of constructing concrete bridges. Its main advantage is the savings made in materials, scaffolding costs and formwork, and above all, the possibility of building a span in many places at the same time. The latter, and especially the cyclicality of concreting individual segments, shortens the time of construction. Cantilever concreting technology in bridges is effective when a span length is between 50 and 250 m.

The typical feature of these bridges is their external appearance, which is shown in Figure

Geometry of Kedzierzyn-Kozle Bridge (Poland)

A characteristic feature of many bridges, as objects of large spans made of prestressed concrete, are their excessive deflections [

During the construction of a bridge, detailed calculations of the deflection of the cantilever span are carried out in order to obtain an appropriate grade line of a bridge object. These calculations include the self-weight of a structure and the effects of assembly prestressing. Technological loads are also important in the case of the deflection of a cantilever. The deformation analysis of the cantilever takes into account the intensive rheological processes occurring in concrete, and also the current climatic conditions. Such calculation results are necessary in order to control the assembly elevation of the entire facility, and in particular, of each segment being built. For the above-mentioned reasons, individual calculations are made, as well as a forecast of the bridge’s grade line for each object.

In this section, a particular phase of construction is analyzed, as shown in Figure

Scheme of a cantilever bridge section

All static and geometric values are functions that are variable in relation to the central axis of box cross-section

The prestressing force is obtained from condition (_{x}_{g}

After considering

Because formula (_{x}_{v}

The deflection of the mid-point of span

In formula (_{0}, the volume weight of concrete _{0}, as well as the assembly loads, are not taken into account. Therefore, the

Due to the fact that the functions in formulas (

In the compliance matrix there are moments of inertia _{v}

The central expression of matrix

Therefore, the expression of vector _{o}_{i}

The bending moments refer to nodes. Geometric characteristics _{g}

The effectiveness of the proposed indicator is illustrated using examples of the analysis of bridges built in Poland – the results of calculations are presented in Table _{v}

Geometric characteristics of cantilever concreting bridges built in Poland

No | Location | |||
---|---|---|---|---|

1 | Grudziądz | 1174 | 0.848 | 180 |

2 | Płock - projekt | 1297 | 0.937 | 148.6* |

3 | Brzeg Dolny | 1189 | 0.859 | 140 |

4 | Kędzierzyn-Koźle | 1796 | 1.297 | 140 |

5 | Kraków | 1583 | 1.143 | 132 |

6 | Łany k. Wrocławia | 1187 | 0.857 | 120 |

7 | Milówka k. Wisły | 1059 | 0.765 | 82 |

For calculating the value of _{v} in Table _{o}^{3}, ^{2} – were assumed. The comparison of _{v} values are similar to those found in constructed buildings.

The deflection _{v} that was determined from formula (

The measurements of grade line changes of bridges’ spans made using the cantilever concreting method have been carried out for many years [

Scheme of Støvset Bridge and the increase of deflections during its operation [

A negative example of the reduction of an excessive deflection can be seen in the Koror-Babelthaupt Bridge with the span of

Although concrete creeping tests have been carried out throughout the 20^{th} century until now, the problem of large deflections of prestressed concrete bridges is still not solved. Therefore, it can be concluded that rheological processes do not reach a finite value during 100 years of bridge operation [

Maximum deflections of the Støvset Bridge span as a function of time [

The waveforms of the deflections of the spans that were built using cantilever concreting technology can be shown in three time ranges [

The effect of changes in the bridge’s grade line that come from the results of measurements of ordinates _{p}_{k}_{p}_{k}

This deflection is not caused by moving loads, but instead by the dead loads of the bridge: the self-weight of the structure and equipment, as well as prestressing. During the deflection of the span, the modulus of the concrete’s deformation

If homogeneity of the concrete in the segments is assumed, then the bending stiffness _{x}

From the form of solution (

In formula (

For this purpose, the measurement of the grade line and deflections at regularly located points along the length of the assessed element with the value c, which were calculated from (_{j},t) is a derivative in the analyzed point j. The mathematical comparison of both values shows the approximation of their values when the c-section tends to zero. However, in the case of measurements on an object, the differences in the value w in points

Figure

Changes in the curvature of the Støvset bridge span in the analyzed time periods

An algorithm of re-calculating the deflection using the Mohr relationship, which is used in structural mechanics for modified bar systems, is proposed in the paper in order to improve (smooth out) the curve of function

In formula (_{j}_{j}

In the case of using the curvature that is determined from the measurements and formula (

In this approach, when the beam line is divided into a sequence of segments with length ^{T} and _{j} from the curvatures and functions found in (_{j}

From formula (_{j} than from the measurements is obtained. This is due to the use of

Changes in the curvature of Støvset Bridge during the smoothing process

Changes in the deflection of Støvset Bridge span during the smoothing process

It is important in the iteration process that the deflection functions do not differ significantly from the initial form that was obtained from the measurements. However, it is assumed that the measurement is carried out correctly and does not contain an erroneous reading. Figures

In the interpretation of the results it is necessary to realize that the result of calculations in the form of function

The general relationship between unit strains

In this paper, a transformed formula (

The process of a change in

Figure _{t} with the values given on the horizontal axis. Therefore, the ordinates from the diagrams in Figure

Changes in curvature and bending moments as a function of creep

In the initial situation, the curvature of the beam that is subjected to bending with moment _{p}

In the final phase of the measurements, according to formula (

The change in curvature _{k} – _{p} that was created in the time interval _{k} < _{p} based on the change of the span’s grade line included in formula (

The bending moment is obtained from equation (

In formulas (_{p}_{k}

Moment function _{i}

Figures

Deflections of the Kedzierzyn-Kozle Bridge span

Curvature of the span of Kedzierzyn-Kozle Bridge during the smoothing process

Figure

Stresses on the top (G) and bottom (D) edges of the bridge span from Figure

The second component of formula (_{ϕ}^{2} was adopted from formula (

A characteristic feature of bridges as large span objects made using cantilever concreting technology is the formation of excessive deflections (

The surveying measurements result in the characteristic functions of the span’s deflection with a parabolic shape. Due to the accuracy of surveying measurements, it is not possible to calculate the internal forces using derivatives of function

A separate issue in concrete cantilever bridges is the construction phase. Its feature is the large dispersion of measurement results of deflection, which is caused by many factors with random characteristics, such as: construction technology, construction time, concreting time, climate, concrete strength, used aggregate, reinforcement grade, prestressing ratio, and the most important – rheological processes. Therefore, calculations for the construction phase must be conducted individually. The paper proposes a general deflection coefficient (