The structural optimization method of steel cantilever used in concrete box girder bridge widening is illustrated in this paper. The structural optimization method of steel cantilever incorporates the conceptual layout design of steel cantilever beam based on the topological theory and the determination of the optimal location of the transverse external prestressed tendons which connect the steel cantilever and the box girder. The optimal design theory and the analysis process are illustrated. The mechanical model for the prestressed steel cantilever is built and the analytical expression of the optimal position of the transverse external tendon is deduced. At last the effectiveness of this method is demonstrated by the design of steel cantilevers which are used to widen an existing bridge.
Structural optimization is an important tool for structural designers because it allows the designers to tailor a structure to a specific performance level required by the owner. Structural optimization is nowadays common in mechanical and aeronautical engineering, and in recent years, it has been progressively adopted for structural engineering and bridges [
In this paper, the structural optimization method of steel cantilever which is used in concrete box girder bridge widening is illustrated. Steel cantilever widening concrete box girder method is a new box girder widening method without piers, which has many advantages, such as shorter construction period, open clearance of span, lesser traffic interference, better traffic capacity, and better economic benefit [
Sectional drawing of original box girder widened by steel cantilevers.
Plane figure of original box girder widened by steel cantilevers.
Diagram of interface between steel and concrete.
The widened box girder is a special structure of steel and concrete combined transversely. Ensuring the reasonable stress on interface between the steel cantilever and the box girder is an important precondition to guarantee that the entire structures work together. Besides, the beautiful shape and reasonable function holes which are used to settle pipelines are essential to the steel cantilevers. The reasonable design of the steel cantilevers is the key problem. Although the conceptual design of the steel cantilevers depends on the designer’s intuition and ability to recognize the role of steel cantilevers in transferring weight and loads to the original box girder, currently, structural optimization may help the designer find the most suitable shape and layout of a steel cantilever from a structural and an architectural point of view [
The topic of this paper is a structural optimization problem which incorporates the conceptual layout design of the steel cantilever and the determination of the optimal location of the transverse external prestressed tendons. The author applied the topological optimization theory to the shape and layout design of steel cantilever and builds the mechanical model for the prestressed steel cantilever. Then the analytical expression of the reasonable acting position of the transverse external prestressed tendons on the steel cantilever is deduced and the steel cantilever structural optimization scheme is proposed.
There are two key issues for optimization design of the steel cantilever.
The first one is the conceptual layout design of steel cantilever beam. The steel cantilever is newly added to the original box girder; the additional live load and dead load on the steel cantilever should be effectively transmitted to the original box girder. This requires the shape, layout of steel cantilever, and the setting of function holes for the pipelines should not damage the structure stiffness; namely, the load path of steel cantilever should not be broken.
The second issue is the decision of the optimal location of the transverse prestressed tendons. The transverse external prestressed tendons which connected the original box girder and the steel cantilevers are key components of the widened structure, whose location influences the stress on the steelconcrete interface as shown in Figure
The aim of topological optimization is to find a conceptual layout of steel cantilever by distributing a given amount of material in a domain, thereby achieving the lightest and stiffest structure while satisfying certain specified design constraints.
Many innovative optimization methods and algorithms have been developed and reported [
Steelconcrete interface is the key position for this composite structure. While pursuing optimization design of the entire steel cantilever beam to make it light, convenient to be processed, and wellformed, it should be premised on that the vehicle load and dead load on steel cantilevers could be transmitted to the interface effectively. The dead load here is the weight of orthotropic steel bridge deck and the bridge deck pavement including concrete paving layer and asphalt concrete paving layer.
In order to get the optimal transmitting path of load on the steel cantilever from which to the interface, the interface of steel cantilever is considered as consolidated from the perspective of model simplification. Because the orthotropic bridge deck slab should be set at top of steel cantilever actually, Ushaped slots need to be reserved for placing bridge deck slab, as it is shown in Figure
Actual structure for topological optimization.
The load on steel cantilever involves dead load and motor vehicle wheel load. The uniform force “
Topological optimization structural under Load Case 1.
The position of wheel load is random and not fixed within the range of the motor vehicle possible passing. Thus, besides the above most unfavorable loading case, wheel load also has other various loading cases. In order to make optimization results widely suitable for various loading conditions, wheel load is hereby fully distributed within the range that wheel load may appear conservatively, which is Load Case 2 as shown in Figure
Topological optimization structure under Load Case 2.
In this paper, minimum compliance of the system is regarded as objective function of the structure optimization. In similar design area, when boundary conditions and load are certain, the smaller the compliance is, the larger the stiffness is [
Suppose external force exerted on the system as
In the equation,
Equilibrium equation of the system is expressed as
Compliance of the system could be expressed as
Mathematical model for topological optimization of steel cantilever structure is established as
As it is illustrated in Section
Considering two load cases, the topology optimization problem to minimize the compliance of the steel cantilever structure while it is subjected to a limited amount of material in the design domain can be written as
In the equation, relative density of the element
In this paper, the optimization analysis is performed using the topological optimization module in ANSYS.
In design domain of steel cantilever, function holes which are used to settle pipelines should be placed in the area that does not need to arrange materials. Leading the obtained optimization results into cartographic software, the shape and function holes of the steel cantilever that do not damage the structure stiffness could be designed according to material distribution results, which means the shape and layout of the function holes design are designed on the premise of ensuring not to damage load transmission path and maintaining the structure stiffness. The specific design procedure will be described later in the application of an actual bridge example.
While analyzing the stress on the interface between steel cantilever and concrete postpouring diaphragm, we suppose that there is no elastic deformation caused to steel cantilever, but only rigid body moves and rotates. The counterforce on concrete interface is in straightline distribution, as it is shown in Figures
Mechanical model one of steelconcrete interface.
Mechanical model two of steelconcrete interface.
The key issue for ensuring that the special composite structures work cooperatively refers to the fact that newly added steel cantilever should contact with concrete box girder closely and the concrete at interface will not be crushed, which means tensile stress will not happen to the interface and compressive stress should not exceed the compressive strength of the concrete at interface.
Combined with actual project, from construction stage to application stage, the stress at steelconcrete interface involves two critical cases. Critical Case I is in construction stage. When steel cantilever is well installed, transverse prestressed tendons are stretched, while orthotropic bridge deck slab and bridge deck pavement are not constructed. During this period, the prestress load takes the main role. The compressive stress on top edge of steelconcrete interface is maximum and the stress on bottom edge of interface is minimum. Critical Case II is in the application stage. In this stage, the orthotropic bridge deck slab is installed and bridge deck pavement is ready. When wheel load is located at the most unfavorable loading position (see Figure
Under this case, only the selfweight of steel cantilever and the action of external prestressed tendon are considered. Suppose that any point “
In Figure
From (
In order to ensure good collaboration of the steel cantilever and original box girder, it should meet the requirements that the tensile stress will not happen to bottom edge of interface and compressive stress at top edge will not exceed compressive strength [
With the above safety requirements, (
From
From
Then, from (
Joining (
From (
From (
Joining (
Simplify the dead load on steel cantilever and wheel load as “
From Figure
From equilibrium of friction along
From (
In order to ensure good collaboration of steel cantilever and original box girder in Critical Case II, it should meet the requirements that tensile stress will not happen to top edge of interface and compressive stress on the bottom edge of interface will not exceed compressive strength of the concrete at interface. Thus, it needs to meet
Referring to the derivation process of Critical Case I, lower limit of “
Combining (
Reasonable values of “
Integrating the topological optimization design for the boundary and layout of steel cantilever and the theoretical derivation result for optimal position of transverse prestressed tendon, the structural optimization scheme of steel cantilever used in concrete box girder widening is proposed. First, make topological optimization analysis on the setting area and design the boundary and layout of steel cantilever beam. Second, based on the optimized shape and combining two critical cases, value range of transverse external prestress and reasonable action range of transverse prestressed tendon could be deduced. Third, select proper value of “
In Figure
Standard cross section of Dalian northeast road overpass (unit: mm).
As it is illustrated in Section
The inclination angle of interface “
As referred to in Section
Topological optimization result for load weight coefficient (1, 0).
Topological optimization result for load weight coefficient (0.5, 0.5).
Topological optimization result for load weight coefficient (0, 1).
Steel cantilever optimization appearance design (unit: cm).
Based on the shape and size of steel cantilevers that have been confirmed, under combined action of dead load, the most unfavorable live load and prestress, the mechanical model for steel cantilever beam is established as in Figure
Steel cantilever mechanics parameter.
Load case 



Cos 
[ 



Dead load + live load  555.43  1430  252.92  0.25  22.4  1.226  Lower limit 0.19 


Selfweight  181.95  1430  85.98  0.25  22.4  1.226  Upper limit 0.32 
Steel cantilever mechanical model (unit: cm).
According to (
Based on above factors and actual conditions of this girder, the position of transverse external prestressed tendons is determined as shown in Figure
Steel cantilever elevational drawing (unit: mm).
Stiffening rib of steel cantilever should be designed in detail on the basis of confirmed shape of steel cantilever and load position of transverse prestressed tendons. During the designing process, the reasonable stress on interface is an important reference index. In order to ensure that the steel cantilever and original girder are reliably combined, principles of the stress on the interface between steel cantilever beam and concrete postpouring diaphragm should be confirmed as follows: the key regions of concrete postpouring diaphragm interface which are under the base plate at web, roof, and bottom flange plate as shown in Figure
Under direction of the above principles, stiffening ribs of steel cantilever are designed in detail. In order to make the force at steel cantilever able to transmit to original girder uniformly, a steel base plate with thickness of 20 mm is placed at the interface between steel cantilever and concrete diaphragm and several stiffening ribs are set on the base plate. Setting stiffening ribs could also guarantee that the external force is transmitted to the interface effectively and evenly. Transverse prestressed tendons especially at both sides of web need to be anchored to Anchor Plate C that is vertical to the webs and the prestress is transferred to interface through Diagonal Rib A, Diagonal Rib B, and Web ② as shown in Figure
Section schematic diagram (unit: mm).
Details of the steel cantilever beam.
A case for optimization design of bridge.
After detailed design, the reasonability and feasibility of the whole optimal design should be verified by analyzing the stress on the steelconcrete interface.
According to above design parameters, finite element model is established and the girder with 3 m long is taken from the widened girder to analyze. In this paper, ANSYS FEA software is used to analyze the stress on the interface. SOLID95 with 20 nodes is used to simulate original concrete box girder and postpouring concrete diaphragm. SHELL63 is used to simulate steel plate structures including steel cantilever beam, orthotropic bridge deck slab, stiffening rib, and steel base plate. Link 10 tensiononly element is used to simulate prestressed tendons. Concrete element at the interface is divided into hexahedral mesh grids and the steel base plate of steel cantilever is divided into the same mesh to ensure that the interface is connected truly and reliably. Link 10 compressiononly element is used to connect the steel base plate and the concrete, which can simulate axial force in the direction of
Finite element model.
Finite element model of interface.
Local stress analysis on widened bridge structure is loaded by
Two critical cases are adopted as follows.
Critical Case I is in the construction state. In this stage the selfweight and prestress are considered. The objective of Case I is checking if compressive stress on top of the interface exceeds allowable value of C50 and if the bottom of the interface open when prestressed tendon is stretched.
Critical Case II is in the service status. Selfweight, dead load, prestress, and live load of four lanes are considered in this stage. The objective is checking if top of the interface between concrete and steel beam opens and if the compressive stress on the bottom exceeds admissible value of C50. Load arrangement should be referred to in Figure
Load spread schematic diagram of model two.
Stress on the interface for Case I is shown in Figure
Concrete interface normal stress diagram of Case I.
Stress on interface for Case II should be referred to in Figure
Concrete interface normal stress diagram of Case II.
Thus, it could be concluded that, under both of the two critical cases, the interface between steel cantilever beam and concrete postpouring diaphragm is ensured to be closed and the concrete at interface is ensured not to be crushed. The stress results show the interface is reasonably compressed, which meets the design requirements.
The structural optimization method of steel cantilever used in concrete box girder bridge widening is illustrated in this paper, which is a new box girder widening method with various advantages. In order to promote actual application of this method, relevant researches on the structural optimization of steel cantilever are made and the following could be concluded.
The authors have introduced the topological optimization theory to get reasonable material distribution results within design area for steel cantilever. And this topological result provides theoretical basis for the determination of shape and arrangement of function holes of steel cantilever beam.
Authors have made stress analysis on the interface between steel cantilever and concrete postpouring diaphragm. In order to prevent tensile stress from happening to the interface under any load cases and make compressive stress not to exceed admissible value for compressive strength of the postpouring diaphragm, basic mechanical model for steel cantilever beam under two critical load cases has been established and the analytical expression of the optimal action range of transverse prestressed tendons has been deduced according to plane crosssection assumption.
In this paper, an optimization design scheme based on the stress at steelconcrete interface has been given, which is applied to a real bridge. The analysis results indicate that using this optimization design scheme to make optimization design on steel cantilever could get the scheme that meets design requirements with reasonable stress. This method could reduce unnecessary trials and save computing resource to the greatest extent, so as to realize rapid and accurate design.
This structural optimization method provides the theoretical support to the promotion of steel cantilever widening concrete box girder method and also promotes the application of structural optimization theory in bridge design.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research described in this paper was financially supported by the science and technology funds of Liaoning Education Department (20131021) and the National Natural Science Foundation of China (51308090).