# Balanced Cantilever Bridges

Solution

## 1. What is a Balanced Cantilever Bridge?

If you are a bridge engineer, you may have heard about the Free Cantilever Method or the Balanced Cantilever Bridge. Then, what is a balanced cantilever bridge? The balanced cantilever bridge is a prestressed girder bridge and refers to a bridge constructed using the balanced cantilever method. One characteristic is that the superstructure is mainly a prestressed box girder.

• Fig. The bridge under construction using the balanced cantilever method

## 2. What is the Balanced Cantilever Method?

The Balanced Cantilever Method refers to a construction method that does not install scaffolding systems under the bridge and completes the superstructure of the bridge by sequentially joining the segments to form a span by post-tensioning and balancing them left and right from each pier using special erection equipment. The balanced cantilever method can be largely classified as Cast-in-Place Cantilever Method and Precast Cantilever Method.

### A. Overview of Construction Phase

The construction sequence using the balanced cantilever method will vary depending on the site conditions, but the overall construction phase is shown in the figure below.

• Fig. Construction sequence of a prestressed concrete bridge using the balanced cantilever method

### B. Overview of Structural System

The structural system of a bridge completed using the balanced cantilever method can be classified into a rigid frame bridge and a supported bridge.

#### B-1. Rigid Frame Bridge Type

The rigid frame bridge is a bridge that rigidly connects the superstructure and substructure and it can be subdivided into: Partially Articulated Bridge, Articulated Bridge and Continuous Bridge. The most common advantage of a rigid frame bridge is that the superstructure and substructure are connected, so no supports are required. In case of continuous structures, the expansion joint system is reduced, which improves mobility and facilitates maintenance.

In addition, in case of a multi-span bridge, it can be designed so that lateral forces disperse to each pier during an earthquake, and the bending moment acting at the bottom of the pier is also reduced, which makes it likely advantageous for earthquakes. Furthermore, since it is a statically indeterminate structure, even if part of a member yields, the stress is redistributed, so sudden failure of the entire structural system can be prevented. Moreover, temporary bridge supports are not needed to resist the unbalance moment at the time of construction. However, since it is a statically indeterminate structure, it is highly influenced by prestress, temperature, concrete shrinkage, creep, and differential settlement of foundations.

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• Fig. Rigid frame bridge type

##### Partially Articulated Balanced Cantilever Bridge

A bridge type constructed using the cantilever method in which the central joint is treated as a hinge. The advantage of this structural type is that it is a statically determinate structure, so structural analysis and design are simple. Also, since the bending moment during construction and the bending moment after construction are the same, the tendon arrangement is simple and less tendons are used. Furthermore, if the foundations have differential settlement, it can be applied to soft grounds because the effect of the settlement is less than that of continuous girder bridge types.

Looking at the disadvantages, since it is considered a statically determinate structure, moment redistribution is not performed, so it has a smaller load carrying capacity compared to a continuous bridge. The central hinge is difficult to design and install, and in the long run, various problem occur in the central hinge. It could increase maintenance costs. Furthermore, since the center of the span is a hinged structure, there is a risk that the girder on the abutment side can be lifted. Lastly, the hinge is less binding to the vertical deflection or deformation angle caused by the creep of the concrete, resulting in large long-term deformations causing problems in driving performance and aesthetics. Therefore, due to these problems, this method is no longer used.

• Fig. Partially articulated balanced cantilever bridge: central hinge type

##### Articulated Balanced Cantilever Bridge

Continuous bridges are not desirable in areas where differential settlement between points is likely to occur due to foundation settlement, etc. Considering these topographical conditions, a method designed so that the difference in slope deflection is less than that of a partially articulated bridge is when an independent Gerber is hung in the middle of a span. However, due to its structural characteristics, it is only applicable for three-span bridges where the span on both sides of the abutment is very short. Since the Gerber beam, which is hung in the center, only receives positive moment, box sections are not necessarily used. Moreover, the self-weight can be reduced by using I-type or T-type sections that match the same number of cantilevers and stabilizing webs. Furthermore, it can be planned as a steel hybrid structure by using steel instead of concrete. However, this bridge type, just like the partially articulated bridge, has the disadvantage of having a small load carrying capacity and increasing expansion joints.

• Fig. Articulated balanced cantilever bridge: gerber type

##### Continuous Balanced Cantilever Bridge

In a continuous balanced cantilever bridge, after the construction of the cantilever, the closure segment is connected to the cast-in-place or precast segment, and then tension is introduced to integrate all girder segments together. This bridge type is easy to maintain because the supports for the bridge piers are not needed, and mobility is good because the expansion joints are reduced. Since it is a high-order statically indeterminate structure, it is highly resistant against earthquakes and rough winds. This structural system is a high-order statically indeterminate structure compared to regular continuous girder bridges, and thus is greatly influenced by concrete dry shrinkage, creep, and differential settlement of foundations. Furthermore, it has a great disadvantage in that the foundations of the bridge are always under lateral loading. However, it can be more economical than continuous beam girders bridges because the piers of a bridge can be designed so that the lateral loads, such as earthquake loads, are distributed evenly amongst the piers, reducing the cost of the bridge supports. Multi-span continuous balanced cantilever bridge is suitable for bridge structures with piers that have high-plasticity (or high-ductility) due to the large expansion of girders and the increase in the deformation of piers due to the effects of concrete creep, drying shrinkage, prestress, and temperature change.

• Fig. Conceptual drawing

• Fig. Construction of a closure segment

• Fig. Continuous balanced cantilever bridge

#### B-2. Supported Bridge Type

A supported bridge is a type of bridge in which box girders are continuous for two or more spans, and the box girders are supported by supports installed on piers and abutments. When supported bridges are constructed using the cantilever method, there are disadvantages such as the need to additionally install temporary fixing devices to resist unbalanced moments generated during construction and constantly getting maintenance checks for the supports of the bridge during its public use. Depending on the type and function of the support, it can be classified as follows:

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• Fig. Supported bridge type

##### Single Fixed Support

The single fixed support method is a type of continuous girder bridge with only one fixed support installed and all other supports are considered movable supports. Since the lateral force of the superstructure due to temperature, seismic loads, etc., is concentrated only on the pier where the fixed support is installed, the cross section of the pier should have a larger cross-sectional area. Furthermore, statically indeterminate forces due to longitudinal temperature changes, creep, and drying shrinkage do not occur.

• Fig. Conceptual diagram of single fixed supported balanced cantilever bridge

##### Multi Fixed Supports

The multi fixed supports method is a type of continuous girder bridge that distributes horizontal forces caused by seismic loads to a number of piers by installing multiple fixed supports. The cross sectional area of the pier on which the fixed support is installed may be smaller than that of the single fixed method. However, statically indeterminate forces occur due to temperature changes, creep and shrinkage. The effects of temperature changes are interpreted in consideration of the ductility of the piers or in consideration of ground deformation.

• Fig. Conceptual drawing of a multi fixed supported balanced cantilever bridge

##### Stopper Method

The stopper method is a type of continuous girder bridge in which one or several piers with fixed supports are installed, and additional stoppers or dampers are installed on piers that have movable supports installed. Excessive horizontal forces due to loads such as seismic loads are distributed to the piers by the stoppers, and expansion of box girders by temperature changes are not restricted. There are three major methods to use stoppers: liquid stoppers, leaf spring stoppers, and a mechanical method using hydraulic cylinders. When seismic design is not considered, the horizontal force is mainly due to temperature loads and vehicle's braking loads, and supports can be designed for this amount. But when the seismic design is considered, the horizontal force may become too excessive to be covered by the support, which can be effectively addressed using stoppers or seismic isolation supports.

• Fig. Stopper method

### C. Superstructure

The superstructure of a balanced cantilever bridge is the prestressed concrete girder. Let us talk about the characteristics and plans for the prestressed concrete girder in bridges.

#### C-1. Girders and Their Manufacturing Method

The girder of a balanced cantilever bridge mainly forms a box girder. This section will explain the historical and engineering content on why box girders are used. Concrete box girders are commonly used in the western part of the United States. Initially, this girder type was made up of only Reinforced Concrete and box girders used existing shapes such as I, T, etc. The cross sectional stiffness is excellent compared to other types of girders, so it can be used for bridges that have long spans. It is also advantageous for the torsion due to its cross-sectional characteristics. As the prestressed concrete became popular, it started to be applied to box girders, which are still used because tendons are easy to install on the upper and lower slabs of the box girder. Box girders have 1 cell or multi-cell structures. For 1 cell structures, the web thickness can be made thick, making it easy to install tendons. Multi cell structures allow for a wider bridge planning compared to 1 cell structures. Each structure has its own pros and cons, so designers and engineers should make appropriate choices according to their plans. The girder used for the balanced cantilever bridge is made into segments, and each segment is attached one by one starting from the pier. Girders can be made directly on site using formwork, or use precast girders produced in girder factories.

• Fig. Cross-section of box girders

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• Fig. In-situ concrete girder (left); Precast girder(right)

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##### In situ Concrete Method

The in-situ concrete method is a construction method of making girders using formwork at a construction site. The girder is made and constructed one segment at a time using a form traveller. Construction work is relatively fast because all work is performed identically and repeatedly. Furthermore, since errors can be corrected for each construction stage, it can increase the precision of construction.

• Fig. In situ concrete method

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##### Precast Concrete Method

The precast concrete method is a construction method of manufacturing the girders in a controlled environment other than the construction site and bringing it to the site for assembly only. Because the precast concrete is made in controlled environments like a manufacturing facility, it is easy to manage, which is an advantage because it can significantly improve the quality of the girder. In addition, since the girder segment can be manufactured parallel with the substructure, the construction period can be shortened compared to the on-site method. When constructing a superstructure, the concrete has already reached considerable age, so creep and drying shrinkage are relatively small, so there is an advantage that the reduction of prestressing force is small. However, the transportation of precast segments is an important issue depending on the site conditions. Furthermore, precise construction management is required, and correction in the event of errors is more difficult than on-site construction methods.

• Fig. Precast Method

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#### C-2. Segment Division

In the Balance Cantilever method, the size of the segment affects the construction period and work efficiency. The size of the segment depends on the capacity of the erection equipment. The length of the segment ranges from 3m to 5m. For segment division, the segment length and weight must be kept constant. If the segment weight is kept constant, the capacity of the erection equipment can be constant as well, thus reducing the cost of the erection equipment. However, since the lengths of the segments are different, the efficiency of rebar work decreases. Furthermore, the number of segments can increase, which can affect the construction time. If the length of the segment is kept constant, there is a disadvantage of increasing the capacity of the erection equipment because the segments of the pier hear are heavy; however, because the length of the segment is constant, construction time is shortened because efficiency of rebar work increases.

• Fig. Plan of segment division

#### C-3. Tendons in Box Girder

The tendons of the balanced cantilever bridge can be installed in the longitudinal and transverse directions. In the longitudinal direction, tendons are located at the upper and lower slabs of the box girder. The tendons in the upper slab resist the negative moments while the tendons in the lower slab resist the positive moments. In the transverse direction, the tendons are installed to resist the moments created by cantilever beams of the box girder. If the longitudinal tendons are not inclined, vertical web tendons may be installed in the web to reinforce shear strength.

• Fig. Tendon arrangement cross-section view

• Fig. Tendon arrangement longitudinal view

### D. Substructure

The types of piers that can be used for balanced cantilever bridges are largely divided into short/stocky column, long/slender column, and twin leaf, twin columns. The short column type is relatively simple to design considering only the strength of concrete and yield of reinforcement without considering the effect of the length of the column or buckling. The long column type can be used to overcome surrounding environments, such as having to secure sufficient space under a bridge or having to pass through a high valley. The long column type requires a more detailed design than the short column type. The reason is that the length of the column is long, causing an effect of the second moment according to the p-delta effect and buckling. Finally, in the twin leaf type, two columns support the girder of the pier head. It is more stable as the two columns distribute the vertical loads. Furthermore, since it is flexible in the horizontal direction, it can respond well to the expansion of continuous bridges. However, due to the thin sections of the piers, the rigidity is small, so a stability check is required, and the design must also be considered for the effects of buckling.

• Fig. Short/Stocky column

• Fig. Long/Slender column

• Fig. Twin leaf/Twin column

## 3. FEM Software Analysis for Balanced Cantilever Bridge

A balanced cantilever bridge is heavily influenced by creep and drying shrinkage, which are time-dependent characteristics of concrete, and can have large stress redistribution phenomenon according to changes in the structural system. Therefore, analysis of the completion stage and construction stage considering the time-dependent material is essential. Designers and engineers need to understand the following four things to carry out the analysis:

### A. Modeling

The longitudinal balanced cantilever bridge can be modeled using the beam element. The length of the beam elements for girders is planned considering the length of the planned segment. If the tapered section property is available in the structural analysis software being used, assign it to the beam element considering the varying cross section of the girder.

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• Fig. Element models of a balanced cantilever bridge

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The design loads for the balanced cantilever bridge design shall be those specified in the design standards for each country. The main loads that must be considered are as follows:

###### 8. Others

In the completion stage, additional reviews are carried for superimposed loads, moving loads, and seismic loads. Most of the loads can be applied using the basic load input functions such as nodal loads, beam element loads, provided by the program. Furthermore, loads can be conveniently considered if there is a separate special function for the program.

### C. Construction Stage Analysis of Balanced Cantilever Bridges

Balanced cantilever bridges have certain characteristics in which loads and boundary conditions change for each construction stage. Also, the properties of materials that change over time must be also considered. Therefore, the safety of the structure must be reviewed through the construction analysis. If a structural analysis software being used supports construction stage analysis, the construction analysis can be carried out conveniently.

• Fig. Construction stage plan of balanced cantilever bridges

• Fig. Element models of the construction stage

#### C-1. Check Lists

Here are some important points to consider when designing a balanced cantilever bridge.

##### Tendon Loss

Post-tensioning is used for balanced cantilever bridges. Prestressing force is applied at each stage of construction. Therefore, calculations are needed to compensate for the loss in the prestressing force that may occur depending on the construction stage. There are two main factors that cause loss in the prestressing force. One is instantaneous loss/Immediate loss, and the other is time dependent loss/long-term prestress loss.

The causes of instantaneous loss/immediate loss include elastic shortening of concrete, slip of anchorage, and friction between tendons and sheaths. All three occur immediately when tension is introduced into the structure. In balanced cantilever bridges, a number of joints occur due to the characteristics of girder segments being constructed in stages. At each joint, unpredictable inclination and curvature of the sheath occurs, so it is important to take care of additional losses due to friction.

The causes of time dependent/long-term loss include creep and shrinkage in concrete, and tendon relaxation. Shrinkage of concrete is a phenomenon in which concrete contracts as it hardens. If the in-situ concrete method is considered, concrete shrinkage should be considered in the calculation of the loss of prestressing force. If precast concrete method is used, it is relatively less affected by shrinkage. Creep refers to a phenomenon in which the internal strain increases when a constant load is applied to the structure for a long period of time. This phenomenon occurs in both concrete and steel structures, and when it occurs in steel structures, it is called relaxation. This also causes loss of prestressing force because additional strain occurs after the structure is completed.

• Fig. Prestress losses

##### Camber

The Camber diagram is for construction management and management of the alignment of the completed bridge. To make a camber diagram, it is necessary to calculate the deflection at the construction stage. The following are considered when calculating for deflections:

###### 9. Deflection due to foundation settlements and changes in reaction forces

When the deflection curve calculation is completed, the camber diagram can be calculated by summing the reverse curve of the deflection curve and the linear curve.

• Fig. Deflection curve and camber diagram

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