research article structural optimization of steel
TRANSCRIPT
Research ArticleStructural Optimization of Steel Cantilever Used inConcrete Box Girder Bridge Widening
Qian Wang,1 Wen-liang Qiu,1 and Sheng-li Xu2
1Faculty of Infrastructure Engineering, Bridge Science Research Institute, Dalian University of Technology, Dalian,Liaoning 116024, China2School of Energy and Power Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China
Correspondence should be addressed to Sheng-li Xu; [email protected]
Received 3 April 2015; Revised 11 June 2015; Accepted 21 June 2015
Academic Editor: Zdenek Kala
Copyright ยฉ 2015 Qian Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The structural optimization method of steel cantilever used in concrete box girder bridge widening is illustrated in this paper. Thestructural optimization method of steel cantilever incorporates the conceptual layout design of steel cantilever beam based on thetopological theory and the determination of the optimal location of the transverse external prestressed tendons which connectthe steel cantilever and the box girder. The optimal design theory and the analysis process are illustrated. The mechanical modelfor the prestressed steel cantilever is built and the analytical expression of the optimal position of the transverse external tendonis deduced. At last the effectiveness of this method is demonstrated by the design of steel cantilevers which are used to widen anexisting bridge.
1. Introduction
Structural optimization is an important tool for structuraldesigners because it allows the designers to tailor a structureto a specific performance level required by the owner.Structural optimization is nowadays common in mechanicaland aeronautical engineering, and in recent years, it hasbeen progressively adopted for structural engineering andbridges [1โ10]; particularly in the aspects of generation ofstrut-and-tie patterns for reinforced concrete structures [11โ13] andfiber-reinforcement retrofitting structures [14โ16], thetopology optimization is widely used. Aiming at differentstructures the corresponding optimization schemes shouldbe proposed to help the designers to find structural formsthat not only better exploitmaterial but also give the structuregreater aesthetic value.
In this paper, the structural optimization method ofsteel cantilever which is used in concrete box girder bridgewidening is illustrated. Steel cantileverwidening concrete boxgirder method is a new box girder widening method withoutpiers, which has many advantages, such as shorter construc-tion period, open clearance of span, lesser traffic interference,
better traffic capacity, and better economic benefit [17, 18].According to thismethod, the original bridge deck is widenedby the orthotropic steel decks that are laid on the cantilevers(Figure 1). The cantilevers are connected to the original boxgirder by transverse external prestressed tendons. Pairs ofsteel cantilever beams are installed on both sides of theoriginal box girder at regular distance (๐ฟ, see Figure 2). Asit is shown in Figure 3, postpouring concrete diaphragms areused to connect with the newly added steel cantilever beams.Concrete postpouring diaphragms are placed in pairs andtheir quantities and locations should be consistent with thoseof newly added steel cantilever beams.
The widened box girder is a special structure of steeland concrete combined transversely. Ensuring the reasonablestress on interface between the steel cantilever and thebox girder is an important precondition to guarantee thatthe entire structures work together. Besides, the beautifulshape and reasonable function holes which are used to settlepipelines are essential to the steel cantilevers. The reasonabledesign of the steel cantilevers is the key problem. Althoughthe conceptual design of the steel cantilevers depends on thedesignerโs intuition and ability to recognize the role of steel
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 105024, 14 pageshttp://dx.doi.org/10.1155/2015/105024
2 Mathematical Problems in Engineering
Orthotropic steel
Steel cantilever Orthotropic steel
bridge deck
beam cantilever
prestressed tendonsTransverse external
Slab
Inclined webPostpouring diaphragm concrete box girder
Original reinforcedAnchor box
Galvanized steel pipe
Figure 1: Sectional drawing of original box girder widened by steel cantilevers.
Steel cantilever beam
Original reinforced concrete box girder
Orthotropic steelbridge deck
L
Figure 2: Plane figure of original box girder widened by steel cantilevers.
Transverse external
Steel-concrete interface
Postpouring Original box girder
Steel cantilever beam
diaphragm
prestressed tendons
inclined web
R
๐
Figure 3: Diagram of interface between steel and concrete.
cantilevers in transferring weight and loads to the originalbox girder, currently, structural optimization may help thedesigner find the most suitable shape and layout of a steelcantilever from a structural and an architectural point of view[19, 20].
The topic of this paper is a structural optimization prob-lem which incorporates the conceptual layout design of thesteel cantilever and the determination of the optimal locationof the transverse external prestressed tendons. The authorapplied the topological optimization theory to the shape andlayout design of steel cantilever and builds the mechanicalmodel for the prestressed steel cantilever. Then the analyticalexpression of the reasonable acting position of the transverse
external prestressed tendons on the steel cantilever is deducedand the steel cantilever structural optimization scheme isproposed.
2. The Optimization Problem Statement
There are two key issues for optimization design of the steelcantilever.
The first one is the conceptual layout design of steelcantilever beam. The steel cantilever is newly added to theoriginal box girder; the additional live load and dead load onthe steel cantilever should be effectively transmitted to theoriginal box girder. This requires the shape, layout of steel
Mathematical Problems in Engineering 3
cantilever, and the setting of function holes for the pipelinesshould not damage the structure stiffness; namely, the loadpath of steel cantilever should not be broken.
The second issue is the decision of the optimal locationof the transverse prestressed tendons.The transverse externalprestressed tendons which connected the original box girderand the steel cantilevers are key components of the widenedstructure, whose location influences the stress on the steel-concrete interface as shown in Figure 3 directly. In order toensure that the steel cantilever is connected with originalconcrete box girder closely and the concrete at interface isnot crushed, the optimal location of transverse prestressedtendons is an important prerequisite. The reasonable stressstate of steel-concrete interface is that there should be notensile stress on the key position of interface and the com-pressive stress should not exceed the compressive strength ofthe concrete at interface, which should be obeyed in deducingthe optimal location of the prestressed tendons.
3. Topological Optimization ofthe Steel Cantilever
The aim of topological optimization is to find a conceptuallayout of steel cantilever by distributing a given amount ofmaterial in a domain, thereby achieving the lightest andstiffest structure while satisfying certain specified designconstraints.
Many innovative optimization methods and algorithmshave been developed and reported [21โ29]. In this paper,topological optimization was carried out through the SIMPmethod [30โ33], due to its computational efficiency andconceptual simplicity.
3.1. Numerical Model. Steel-concrete interface is the keyposition for this composite structure. While pursuing opti-mization design of the entire steel cantilever beam to make itlight, convenient to be processed, and well-formed, it shouldbe premised on that the vehicle load and dead load on steelcantilevers could be transmitted to the interface effectively.The dead load here is the weight of orthotropic steel bridgedeck and the bridge deck pavement including concrete pavinglayer and asphalt concrete paving layer.
In order to get the optimal transmitting path of load onthe steel cantilever from which to the interface, the interfaceof steel cantilever is considered as consolidated from theperspective of model simplification. Because the orthotropicbridge deck slab should be set at top of steel cantileveractually, U-shaped slots need to be reserved for placingbridge deck slab, as it is shown in Figure 4. Thus, single-ended consolidated structure with U-shaped slots is the basicstructure for topological optimization of steel cantilever.
The load on steel cantilever involves dead load andmotorvehicle wheel load. The uniform force โ๐
0โ of dead load as
shown in Figure 5 is theweight of orthotropic bridge deck anddeck pavement equally distributed on each steel cantilever.According to the regulations of General Code for Design ofHighway Bridges and Culverts [34], the most unfavorablewheel load distribution is arranged. Settingwidening one lane
Design domain
Undesigned domain
Boundary lines of design domain
๐
Figure 4: Actual structure for topological optimization.
Wheel load Wheel load Wheel load
Dead load
L0 L0
q0
L1
q1 q1 q1
b dw
Figure 5: Topological optimization structural under Load Case 1.
Dead Loadq0
Wheel load q1
b1
Figure 6: Topological optimization structure under Load Case 2.
unilaterally as an example, within widening range of steelcantilever, twowheels and part of onewheelmay be arranged.Wheel uniform load โ๐
1โ could be calculated according to
the vehicle axle load and wheel action range. Optimizationanalysis for Load Case I refers to the combined action of deadload uniform force โ๐
0โ and actual most unfavorable wheel
load, as it is shown in Figure 5.The position of wheel load is random and not fixedwithin
the range of the motor vehicle possible passing.Thus, besidesthe above most unfavorable loading case, wheel load also hasother various loading cases. In order to make optimizationresults widely suitable for various loading conditions, wheelload is hereby fully distributed within the range that wheelload may appear conservatively, which is Load Case 2 asshown in Figure 6. Besides the dead load and wheel load onthe steel cantilever, the self-weight of the steel cantilever alsoshould be considered during the topological optimizationprocess.
3.2. Mathematical Model. In this paper, minimum com-pliance of the system is regarded as objective function of
4 Mathematical Problems in Engineering
the structure optimization. In similar design area, whenboundary conditions and load are certain, the smaller thecompliance is, the larger the stiffness is [33, 35].
Suppose external force exerted on the system as ๐น, thenits strain energy could be expressed as
๐ธ๐ =12โซฮฉ
๐ (๐ข)๐๐ท๐ (๐ข) ๐ฮฉ =
12๐น๐๐. (1)
In the equation,ฮฉ is given design area; ๐(๐ข) is strain underload ๐น; ๐ข is elastic deformation of any point in design areaunder load ๐น; ๐ท is elastic matrix; ๐น is load vector; and ๐ isdisplacement vector.
Equilibrium equation of the system is expressed as
๐พ๐ = ๐น, (2)
wherein ๐พ represents the stiffness matrix of the system and๐พ๐= ๐พ. Joining (1) and (2) together, it could be concluded
that
๐ธ๐ =12๐๐๐พ๐๐ =
12๐๐๐พ๐. (3)
Compliance of the systemcould be expressed as๐ถ = ๐น๐๐.
Compared with (1), it could be concluded that if complianceof the system is minimum, its strain energy is minimum.
Mathematical model for topological optimization of steelcantilever structure is established as
๐ (๐ฅ) = ๐ฅ๐๐0,
๐ธ (๐ฅ) = ๐ฅ๐
๐๐ธ0;
(4)
wherein ๐0 and ๐ธ0 represent density and elastic matrixof cantilever after subdivision, respectively; ๐ฅ
๐represents
relative density of the element; and ๐ represents penaltyfactor.
As it is illustrated in Section 3.1, actual topological struc-ture is mainly designed according to two load cases (see Fig-ures 5 and 6). Making stiffness of steel cantilever the largest,the best material distribution results can be calculated. Inorder to ensure that the steel cantilever can transmit the loadsto the interface effectively, Load Case 1 and Load Case 2are considered together to confirm the boundary and layoutof steel cantilever. The multiload approach [36, 37], whichconsiders multiload cases by using the weight coefficient, canbe used to perform the optimization boundary and layout ofsteel cantilever.
Considering two load cases, the topology optimizationproblem to minimize the compliance of the steel cantileverstructure while it is subjected to a limited amount of materialin the design domain can be written as
Find ๐ = [๐ฅ1, ๐ฅ2, ๐ฅ3, . . . , ๐ฅ๐]๐
Minimize: ๐ถ (๐) = ๐1๐๐
1๐พ๐1 + ๐2๐๐
2๐พ๐2
= ๐1
๐
โ
๐=1(๐ฅ๐)๐๐๐
1๐๐พ๐1๐
+ ๐2
๐
โ
๐=1(๐ฅ๐)๐๐๐
2๐๐พ๐2๐
Subjected to:๐
โ
๐=1๐ฅ๐โค ๐
๐พ๐1 = ๐น1
๐พ๐2 = ๐น2
0 < ๐ฅmin < ๐ฅ๐< 1.
(5)
In the equation, relative density of the element๐ฅ๐is design
variable; ๐ represents volume coefficient; ๐พ๐1 = ๐น1 is theequilibrium equation for Load Case 1, and ๐พ๐2 = ๐น2 is theequilibrium equation for Load Case 2; ๐1 and ๐2 are the weightcoefficients for Load Case 1 and Load Case 2. In this case,themathematicalmodel could be described as finding densitydistribution of each element to make the stiffness of the steelcantilever the largest in certain combination of load weightcoefficients.
In this paper, the optimization analysis is performedusing the topological optimization module in ANSYS.
In design domain of steel cantilever, function holes whichare used to settle pipelines should be placed in the area thatdoes not need to arrange materials. Leading the obtainedoptimization results into cartographic software, the shape andfunction holes of the steel cantilever that do not damage thestructure stiffness could be designed according to materialdistribution results, which means the shape and layout of thefunction holes design are designed on the premise of ensuringnot to damage load transmission path and maintaining thestructure stiffness. The specific design procedure will bedescribed later in the application of an actual bridge example.
4. Determination of Optimal Location ofthe Transverse External Tendon
4.1. Mechanical Model. While analyzing the stress on theinterface between steel cantilever and concrete postpouringdiaphragm, we suppose that there is no elastic deformationcaused to steel cantilever, but only rigid body moves androtates. The counterforce on concrete interface is in straight-line distribution, as it is shown in Figures 7 and 8.
The key issue for ensuring that the special compositestructures work cooperatively refers to the fact that newlyadded steel cantilever should contactwith concrete box girderclosely and the concrete at interface will not be crushed,which means tensile stress will not happen to the interfaceand compressive stress should not exceed the compressivestrength of the concrete at interface.
Combined with actual project, from construction stageto application stage, the stress at steel-concrete interfaceinvolves two critical cases. Critical Case I is in constructionstage. When steel cantilever is well installed, transverseprestressed tendons are stretched, while orthotropic bridgedeck 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 steel-concrete interfaceis maximum and the stress on bottom edge of interface isminimum. Critical Case II is in the application stage. In
Mathematical Problems in Engineering 5
M0
F
P0
O
x
q2
q1
y
๐
x1
x0
Figure 7: Mechanical model one of steel-concrete interface.
this stage, the orthotropic bridge deck slab is installed andbridge deck pavement is ready. When wheel load is locatedat the most unfavorable loading position (see Figure 5), thestress on top edge of steel-concrete interface is minimumand the stress on bottom edge of interface is maximum.Under the above two critical cases, if tensile stress is ensurednot to happen to top edge and bottom edge of interfaceand compressive stress is ensured not to exceed compressivestrength of the concrete at interface, the stress distributionunder other general load cases can be determined to satisfythe structural requirements.
4.2. Theoretical Inference for the Optimal Position ofthe Transverse External Tendon
4.2.1. Stress Analysis on Critical Case I. Under this case, onlythe self-weight of steel cantilever and the action of externalprestressed tendon are considered. Suppose that any pointโ๐โ on the steel cantilever interface is reference point for theforce analysis and the equivalent stress analysis is performedbased on which. Meanwhile, simplify self-weight of steelcantilever beam as โ๐
0โ concentrated force and โ๐
0โ bending
moment that go through the reference center โ๐.โ Transverseexternal prestress is โ๐นโ and the distance between its loadposition and bending center is โ๐ฅ
1.โ Length of interface is โ๐,โ
the distance between top edge of interface and bending centeris โ๐ฅ0,โ and the distance between bottom edge of interface
and bending center is โ๐ โ ๐ฅ0.โ Counterforce on the top edge
of interface is โ๐1โ and counterforce on the bottom edge of
interface is โ๐2.โ As elasticity modulus of steel is much larger
than that of concrete, the stress on the interface is supposed tobe in linear distribution. Mechanical model for Critical CaseI should be referred to in Figure 7.
In Figure 7, it could be concluded that
๐นโ๐0 cos ๐ = โซ
๐โ๐ฅ0
โ๐ฅ0
๐ (๐ฅ) ๐๐ฅ (6)
๐น๐ฅ1 +โซ๐โ๐ฅ0
โ๐ฅ0
๐ (๐ฅ) โ ๐ฅ ๐๐ฅโ๐0 = 0. (7)
From (6), ๐1 and ๐2 can be expressed as
๐1 =2 (๐น โ ๐0 cos ๐)
๐โ ๐2
๐2 =2 (๐น โ ๐0 cos ๐)
๐โ ๐1.
(8)
In order to ensure good collaboration of the steel can-tilever and original box girder, it should meet the require-ments that the tensile stress will not happen to bottom edgeof interface and compressive stress at top edge will not exceedcompressive strength [๐] of the concrete at interface. Thus, itneeds to meet ๐
1โค [๐]๐ก, where โ๐กโ represents the width of the
concrete postpouring diaphragm, and it also needs to meet๐2โฅ 0.With the above safety requirements, (8) can be changed
as
๐1 โค2 (๐น โ ๐0 cos ๐)
๐
๐2 โฅ2 (๐น โ ๐0 cos ๐)
๐โ [๐] โ ๐ก.
(9)
From ๐1 โค [๐] โ ๐ก and ๐1 โค 2(๐น โ๐0 cos ๐)/๐, the followingequation is gotten:
๐1 โค ๐0, (10)
wherein ๐0 = min{[๐] โ ๐ก, 2(๐น โ ๐0 cos ๐)/๐}.From ๐2 โฅ 0 and ๐2 โฅ 2(๐น โ ๐0 cos ๐)/๐ โ [๐] โ ๐ก, the
following equation is gotten:
๐2 โฅ ๐0, (11)
wherein ๐0 = max{0, 2(๐น โ ๐0 cos ๐)/๐ โ [๐] โ ๐ก}.Then, from (7), taking๐ฅ
0= ๐/2, it could be concluded that
๐น๐ฅ1 โ12(๐1 โ ๐2) โ ๐ โ (
2๐3โ๐
2)โ๐0 = 0. (12)
Joining (8) into (12), we can get
๐น๐ฅ1 โ๐2
6[(๐น โ ๐0 cos ๐)
๐โ ๐2] = ๐0 (13)
๐น๐ฅ1 โ๐2
6[๐1 โ
(๐น โ ๐0 cos ๐)๐
] = ๐0. (14)
From (10) and (13), the following equation can be gotten:
๐ฅ1 โค [๐0๐
2โ (๐น โ ๐0 cos ๐) ๐
6+๐0] โ
1๐น. (15)
From (11) and (14), the following equation can be gotten:
๐ฅ1 โค [(๐น โ ๐0 cos ๐) ๐ โ ๐0๐
2
6+๐0] โ
1๐น. (16)
Joining (15) and (16) together, the upper limit of ๐ฅ1is
given by
๐ฅ1 โค min{[๐0๐
2โ (๐น โ ๐0 cos ๐) ๐
6+๐0]
โ 1๐น, [(๐น โ ๐0 cos ๐) ๐ โ ๐0๐
2
6+๐0] โ
1๐น} ,
(17)
6 Mathematical Problems in Engineering
M
F
P
O
x
q2
q1
y
๐
x1
x0
Figure 8: Mechanical model two of steel-concrete interface.
4.2.2. Stress Analysis on Critical Case II. Simplify the deadload on steel cantilever and wheel load as โ๐โ concentratedforce and โ๐โ bendingmoment that go through the referencepoint โ๐.โ The transverse external prestress is โ๐นโ and thedistance between its load position and bending center is โ๐ฅ
1.โ
Length of the interface is โ๐,โ the distance between top edgeof cross section and bending center is โ๐ฅ
0,โ and the distance
between bottom edge of interface and bending center isโ๐ โ ๐ฅ
0.โ Counterforce on top edge of interface is โ๐
1โ and
counterforce on bottom edge of interface is โ๐2.โ The stress
on the interface is also supposed to be in linear distribution.Mechanical model for Critical Case II is shown in Figure 8.
From Figure 8, it could be concluded that
๐นโ๐ cos ๐ = โซ
๐โ๐ฅ0
โ๐ฅ0
๐ (๐ฅ) ๐๐ฅ (18)
๐น๐ฅ1 +โซ๐โ๐ฅ0
โ๐ฅ0
๐ (๐ฅ) โ ๐ฅ ๐๐ฅโ๐ = 0. (19)
From equilibrium of friction along ๐ฅ-axis, it can be gottenthat
๐ โ sin ๐ < ๐ โ โซ
๐โ๐ฅ0
โ๐ฅ0
๐ (๐ฅ) โ ๐๐ฅ. (20)
From (18) and (20), the lower limit of external prestressโ๐นโ could be valued as
๐น > ๐ โ (sin ๐๐
+ cos ๐) . (21)
In order to ensure good collaboration of steel cantileverand original box girder in Critical Case II, it should meet therequirements that tensile stress will not happen to top edgeof interface and compressive stress on the bottom edge ofinterface will not exceed compressive strength of the concreteat interface.Thus, it needs to meet ๐
1โฅ 0 and ๐
2โค [๐]๐ก at the
same time.Referring to the derivation process of Critical Case I,
lower limit of โ๐ฅ1โ could be expressed as
๐ฅ1 โฅ max{[๐โ(๐น โ ๐ cos ๐) ๐ โ ๐๐2
6]
โ 1๐น, [๐โ
๐๐2โ (๐น โ ๐ cos ๐) ๐
6] โ
1๐น} .
(22)
4.2.3. Load Position of Transverse Prestress and Value of Exter-nal Prestress. Combining (17) and (22), theoretical valuerange of โ๐ฅ
1โ and the load position of transverse prestressed
tendon could be expressed as
๐ฅ1 โฅ max
{{{{{
{{{{{
{
[๐ โ(๐น โ ๐ cos ๐) ๐ โ ๐๐2
6] โ
1๐น
[๐ โ๐๐
2โ (๐น โ ๐ cos ๐) ๐
6] โ
1๐น
}}}}}
}}}}}
}
๐ฅ1 โค min
{{{{{
{{{{{
{
[๐0๐
2โ (๐น โ ๐0 cos ๐) ๐
6+๐0] โ
1๐น
[(๐น โ ๐0 cos ๐) ๐ โ ๐0๐
2
6+๐0] โ
1๐น
}}}}}
}}}}}
}
.
(23)
Reasonable values of โ๐นโ and โ๐ฅ1โ are the important
premise to ensure the stress on the interface satisfy therequirements. To sum up, in the steel cantilever wideningmethod, when making design for transverse external pre-stressed tendon, (21) should be referred to, according towhich, limit for transverse external prestress โ๐นโ is provided.Within this range, type and number of prestressed tendonscould be set to confirm the value of โ๐น.โ Then, from (23),value range of โ๐ฅ
1โ could be calculated, namely, theoretical
range for load position of transverse external prestress ten-don. The value of โ๐นโ can be adjusted until the load positionof transverse prestressed tendon is reasonable and practical.
5. Structural Optimization Scheme ofSteel Cantilever
Integrating the topological optimization design for theboundary and layout of steel cantilever and the theoreticalderivation result for optimal position of transverse pre-stressed tendon, the structural optimization scheme of steelcantilever used in concrete box girder widening is proposed.First, make topological optimization analysis on the settingarea and design the boundary and layout of steel cantileverbeam. Second, based on the optimized shape and combiningtwo critical cases, value range of transverse external prestressand reasonable action range of transverse prestressed tendoncould be deduced. Third, select proper value of โ๐นโ to ensurethat position of transverse external prestressed tendons isreasonable and practical and thenmake detail design on stiff-ening rib near interface of steel cantilever beam. Finally,makeaccurate finite element analysis on contact stress at steel-concrete interface under critical case to verify if the valuesof design parameters are reasonable. Steel cantilever couldbe designed according to this optimization analysis scheme,which could effectively reduce trials and save computingresources.
6. Application
6.1. Actual Bridge Example. In Figure 9, it shows the sectionof a two-lane bridge. This bridge is a reinforced concretecontinuous box girder bridge having a width of 9.5m. The
Mathematical Problems in Engineering 7
150 ร 150 150 ร 150
R1200
R500
95001000 10001250
2250 2250
1300
5000160
180
1300
120
1250500 5001800 1800400
Figure 9: Standard cross section of Dalian northeast road overpass (unit: mm).
beam depth is 1.3m; the thicknesses of roof and bottomplate are 18 cm and 16 cm, respectively. And the thicknessesof webs are shown in Figure 9. The results of health surveyindicate that the bridge is in good condition. In order toease traffic pressure of this bridge, it needs to be widenedfrom two lanes to four lanes. This bridge is of an importantgeographical location with intensive underground pipelinesand surrounding buildings and the traffic is not allowed to bestopped. In order to reduce building demolition and preventtraffic confliction on the ground, the steel cantilever wideningconcrete box girdermethodwithout piers is adopted towidenthe bridge.
6.2. Optimization Design for the Shape of Steel Cantilever. Asit is illustrated in Section 3.1, fully distributed dead load anduniform wheel load are the major load case to control theshape design of steel cantilever. For this bridge, pavementlayer of pitch is 8 cm and pavement layer of concrete is10 cm. Thickness of the roof of orthotropic bridge deck slabis 14mm and thickness of U-shaped rib is 6mm. A pair ofsteel cantilever beams are set every 3m, thickness of the webis 14mm, and thickness of bottom plate is 20mm. Combinedwith the above actual loading conditions, dead uniform loadโ๐0โ and wheel uniform load โ๐1โ are calculated as ๐0 =
21.9 kN/m and ๐1 = 140.8 kN/m (axle weight of vehicle is130 kNwhich considers dynamic loadmagnification factor as1.3), respectively. Load arrangement form is the same as in theabove Figures 5 and 6.
The inclination angle of interface โ๐โ should be deter-mined by comprehensively considering the following factors:transverse external prestressed tendons should be vertical tothe interface; the external prestressed tendons which crossthe original box girder should try to minimize the damageof the original structure; bending radius โ๐ โ of transverseexternal prestressed tendons should meet the minimumspecified value which is 1.5m for epoxy coated strands. Withintegration of the above factors, the inclination angle ofinterface here is valued as ๐ = 75.5โ and the bending radiusof transverse prestressed tendons is ๐ = 2m.
As referred to in Section 3.2, the objective function of thetopological optimization problem is to make the stiffness ofthe steel cantilever the largest. Design variable is the relativedensity of the element for the materials within design area. Inorder to ensure that the steel cantilever transmits the loadsto the interface effectively, Load Case 1 and Load Case 2are considered together to confirm the boundary and layout
A
Design boundary of
B DC E
steel cantileverX
Y
Z
MX
Figure 10: Topological optimization result for load weight coeffi-cient (1, 0).
X
Y
Z
A
Design boundary of steel cantilever
CD
EB
MX
Figure 11: Topological optimization result for load weight coeffi-cient (0.5, 0.5).
X
Y
Z
A
Design boundary ofsteel cantilever
B C1
C2
DE
MX
Figure 12: Topological optimization result for load weight coeffi-cient (0, 1).
of steel cantilever. As to this minimum compliance designproblem, the material distribution results for three kinds ofweight coefficients (the weight coefficients of Load Case 1and Load Case 2 are selected as (1, 0), (0.5, 0.5), and (0, 1))are compared. Besides, the optimization results for differentvolume coefficients (20% to 60%) are compared too. Basedon the layout of them, the material distribution of 40% isselected for the more reasonable shape and hole parametersfrom the point of actual engineering.The optimization resultsfor different kinds of weight coefficients when the volumecoefficient is 40% are given in Figures 10 to 12. As can be seenfrom them, on the premise of the same volume coefficient,the optimization results for different weight coefficients aresimilar, wherein blue area represents the area that does notneed to arrange materials during structure design and redareas represent the areas that need to arrange materials. The
8 Mathematical Problems in Engineering
0.6
23.3
41.8
32
2
26 32 26 32 26 32 26 32 32 46.97
2 137.6
1.4
26
t = 1.4
๐ = 75.5โ
Figure 13: Steel cantilever optimization appearance design (unit: cm).
Transverseprestress F
Dead load
100 180 100
60
68.7
4
68.7
4
60
12.62
Wheel loadWheel load Wheel load21.9 kN/m
140.8 kN/m 140.8 kN/m 140.8 kN/m
M(M0)
P(P0)x
y
O
x1
๐ = 75.5โ
Figure 14: Steel cantilever mechanical model (unit: cm).
results of topological optimization are used to confirm theboundary of steel cantilever and the positions of functionholes. As it is shown in Figures 10 to 11, in the domain ofthe boundary of the steel cantilever, function holes whichare used to settle pipelines should be placed in the blue areamarked as A, B, C, and D. The area marked as E should notbe punched because, under the transverse prestressed load,domain E is the key area of the prestress transmitting. Butfor this bridge, lighting at bridge deck is provided by streetlamps on the ground and there are no other requirements forpipelines to pass by in early stage. Thus, functional holes arenot needed to be set in steel cantilever beam.The streamlinedboundary of steel cantilever is extracted as it is shown inFigure 13. In the future, the function holes can be set in theblue area marked as A, B, C, and D when they are needed.
6.3. Design for Load Position of Transverse Prestressed Ten-dons. Based on the shape and size of steel cantileversthat have been confirmed, under combined action of deadload, the most unfavorable live load and prestress, themechanical model for steel cantilever beam is establishedas in Figure 14. The uniform load caused by bridge deckpavement is 21.9 kN/m and uniform force made by singlewheel is 140.8 kN/m, which is Critical Case II as illustrated
in Section 4.1. The dead load on steel cantilever beam andwheel load are simplified as a concentrated force โ๐โ andbending moment โ๐โ that go through the bending centerโ๐.โ The other critical case refers to the fact that onlyself-weight of steel cantilever beam and action of externalprestressed tendons are considered during construction. Self-weight of steel cantilever beam is simplified as a concentratedforce โ๐
0โ and bending moment โ๐
0โ that go through
the bending center. The concrete strength grade of originalgirder is C30. In order to supply sufficient resistance to thesteel cantilever beam, the concrete postpouring diaphragmneeds enough compressive strength. So, C50 is selected forthe postpouring diaphragm. During the derivation process,ultimate compressive strength of concrete [๐] is valued as22.4MPa according to the regulations for bridges โจJTG D62-2004โฉ. Specific values of other parameters under the twocritical cases should be referred to in Table 1.
According to (23), the range of theoretical load position oftransverse prestressed tendons in this example is calculated tobe 0.19m < ๐ฅ1 < 0.32m.Within this range, the specific posi-tion of transverse prestressed tendons should be determinedby considering these factors: when the transverse prestressedtendons go through the original girder, the damage to originalgirder should be minimized and the bending radius โ๐ โ
Mathematical Problems in Engineering 9
Table 1: Steel cantilever mechanics parameter.
Load case ๐ (๐0) (kNm) ๐น (kN) ๐ (๐
0) (kN) Cos ๐ [๐] (MPa) ๐ (m) ๐ฅ
1(m)
Dead load + live load 555.43 1430 252.92 0.25 22.4 1.226 Lower limit 0.19Self-weight 181.95 1430 85.98 0.25 22.4 1.226 Upper limit 0.32
t = 6mm
t = 20mm
t = 20mm
t = 14mm
t = 16mm
t = 14mm
I
I3
4
2
12345
ABCDE
6
1
CA
B
6
DE
5
0
IIIIII
II
IIx1=20.5
R1500
: roof, 14mm: web, 14mm: bottom flange plate, 20mm: U-shaped rib, 6mm: small stiffening rib, 16mm: base plate, 20mm
: prestressed tendon: preembedded steel tube
: upper diagonal rib, 20mm
: anchor plate, 20mm: lower diagonal rib, 20mm
Figure 15: Steel cantilever elevational drawing (unit: mm).
of transverse external prestressed tendons should meet theminimum specified value of epoxy coating steel strands.
Based on above factors and actual conditions of thisgirder, the position of transverse external prestressed tendonsis determined as shown in Figure 15, wherein๐ฅ1 takes 20.5 cmand bending radius of transverse prestressed tendons is ๐ =
2m. The transverse prestress is 1430 kN, which could beoffered by 10๐15.2mm prestressed tendons, which are setsymmetrically on both sides of steel cantilever slab as shownin Figure 17. The steel cantilevers at both sides of box girderare transversely combined with original box girder by 2 bun-dles (each bundles is composed by 5๐15.2mm prestressedtendons) of external prestressed epoxy steel strands.
6.4. Detailed Design of Stiffening Rib. Stiffening rib of steelcantilever should be designed in detail on the basis ofconfirmed shape of steel cantilever and load position of trans-verse prestressed tendons. During the designing process,the reasonable stress on interface is an important referenceindex. In order to ensure that the steel cantilever and originalgirder are reliably combined, principles of the stress onthe interface between steel cantilever beam and concretepostpouring diaphragm should be confirmed as follows: thekey regions of concrete postpouring diaphragm interfacewhich are under the base plate at web, roof, and bottomflange plate as shown in Figure 15 should keep in compressionto avoid from being separated from steel cantilever and themaximum compressive stress of concrete interface should beensured not to exceed local compressive admissible value ofpostpouring diaphragm concrete.
Under direction of the above principles, stiffening ribsof steel cantilever are designed in detail. In order to makethe force at steel cantilever able to transmit to original girderuniformly, a steel base plate with thickness of 20mm isplaced at the interface between steel cantilever and concretediaphragm and several stiffening ribs are set on the base plate.Setting stiffening ribs could also guarantee that the externalforce is transmitted to the interface effectively and evenly.Transverse prestressed tendons especially at both sides of webneed to be anchored to Anchor Plate C that is vertical tothe webs and the prestress is transferred to interface throughDiagonal Rib A, Diagonal Rib B, and Web B as shown inFigure 15. Specific size of various components should bereferred to in Figures 15 and 16. Structure of the actual bridgeshould be referred to in Figure 18.
6.5. Stress Analysis on Steel-Concrete Interface. After detaileddesign, the reasonability and feasibility of the whole optimaldesign should be verified by analyzing the stress on the steel-concrete interface.
According to above design parameters, finite elementmodel is established and the girder with 3m long is takenfrom the widened girder to analyze. In this paper, ANSYSFEA software is used to analyze the stress on the interface.SOLID95 with 20 nodes is used to simulate original concretebox girder and postpouring concrete diaphragm. SHELL63is used to simulate steel plate structures including steelcantilever beam, orthotropic bridge deck slab, stiffening rib,and steel base plate. Link 10 tension-only element is usedto simulate prestressed tendons. Concrete element at theinterface is divided into hexahedral mesh grids and the steel
10 Mathematical Problems in Engineering
III-III cross section
40
6
5
82
3015
15
I-I cross section
1
2
3
40
t = 14
t = 20
II-II cross section
1
2
5
0B
5 68.7
4
20.5
137.
58
68.7
4
3
6
40
x
y
Figure 16: Section schematic diagram (unit: mm).
Transverse prestressingtendons
Anchor boxSlab
Figure 17: Details of the steel cantilever beam.
Figure 18: A case for optimization design of bridge.
base plate of steel cantilever is divided into the same meshto ensure that the interface is connected truly and reliably.Link 10 compression-only element is used to connect the steelbase plate and the concrete, which can simulate axial forcein the direction of ๐ฆ, while the contact conditions in otherdirections are simulated by the coupled equations. Finiteelement model is shown in Figures 19 and 20.
Local stress analysis on widened bridge structure isloaded by 20๐ก vehicle with 70 kN front shaft and 130 kNback shaft. The most unfavorable condition is the case whenthe back shaft is loaded on the cantilever. According to theregulations of bridges โจJTG D62-2004โฉ, impact effect should
be included here and impact coefficient should take 1.3.Thus,the load of back shaft should be exerted as 1.3 ร 130 kN =169 kN.
Two critical cases are adopted as follows.Critical Case I is in the construction state. In this stage
the self-weight and prestress are considered. The objectiveof Case I is checking if compressive stress on top of theinterface exceeds allowable value of C50 and if the bottomof the interface open when prestressed tendon is stretched.
Critical Case II is in the service status. Self-weight, deadload, prestress, and live load of four lanes are considered inthis stage. The objective is checking if top of the interface
Mathematical Problems in Engineering 11
Figure 19: Finite element model.
Concrete interfaceSteel interface
Figure 20: Finite element model of interface.
100 100180 180 180 180225 22510 10130 130
84.5kN 84.5kN 84.5kN 84.5 kN 84.5kN 84.5kN 84.5kN84.5kN
Figure 21: Load spread schematic diagram of model two.
between concrete and steel beam opens and if the compres-sive stress on the bottom exceeds admissible value of C50.Load arrangement should be referred to in Figure 21.
Stress on the interface for Case I is shown in Figure 22. Itcould be concluded from this figure that the key regions ofinterface at web, roof, and bottom flange is under compres-sion. Localmaximumcompressive stress of concrete interfaceis 14.4MPa and this value is smaller than compressivestrength of C50 concrete used for postpouring diaphragm as22.4MPa. And from enlarged stress nephogram for normalstress at bottom edge of the interface, it could be concludedthat tensile stress does not happen to bottom edge of interfaceunder this case.
Stress on interface for Case II should be referred to inFigure 23. It could be concluded that the key regions of theinterface is also under compression. Local maximum com-pressive stress for concrete is 16.9MPa, which is smaller than
compressive stress of concrete postpouring diaphragm, so asto ensure that the concrete at interface is safely compressed.In this case, the top of the interface is easy to open, while theenlarged cloud diagram for the stress at top edge shows thatthis position is under compression too.
Thus, it could be concluded that, under both of the twocritical cases, the interface between steel cantilever beam andconcrete postpouring diaphragm is ensured to be closed andthe concrete at interface is ensured not to be crushed. Thestress results show the interface is reasonably compressed,which meets the design requirements.
7. Conclusions
The structural optimization method of steel cantilever usedin concrete box girder bridge widening is illustrated in thispaper, which is a new box girder widening method with
12 Mathematical Problems in Engineering
Stress nephgram for normal stress at bottom edge of concrete interface
concrete interfaceStress nephgram for normal stress of
โ0.144E + 08
โ0.126E + 08
โ0.107E + 08
โ0.889E + 07
โ0.706E + 07
โ0.523E + 07
โ0.340E + 07
โ0.157E + 07
262259
0.209E + 07
โ0.313E + 07โ0.270E + 07โ0.227E + 07โ0.185E + 07โ0.142E + 07โ992796โ565975โ139155287666714486
Figure 22: Concrete interface normal stress diagram of Case I.
Stress nephgram for normal stress of concrete interface
Stress nephgram for normal stress at topedge of concrete interface
โ0.169E + 08
โ0.147E + 08
โ0.124E + 08
โ0.102E + 08
โ0.802E + 07
โ0.580E + 07
โ0.359E + 07
โ0.138E + 07
834130
0.305E + 07
โ0.776E + 07โ0.682E + 07โ0.588E + 07โ0.493E + 07โ0.399E + 07โ0.305E + 07โ0.211E + 07โ0.117E + 07โ233458706807
Figure 23: Concrete interface normal stress diagram of Case II.
various advantages. In order to promote actual application ofthis method, relevant researches on the structural optimiza-tion of steel cantilever are made and the following could beconcluded.
(1) The authors have introduced the topological opti-mization theory to get reasonable material distribu-tion results within design area for steel cantilever.And this topological result provides theoretical basisfor the determination of shape and arrangement offunction holes of steel cantilever beam.
(2) Authors have made stress analysis on the interfacebetween steel cantilever and concrete postpouringdiaphragm. In order to prevent tensile stress fromhappening to the interface under any load cases andmake compressive stress not to exceed admissiblevalue for compressive strength of the postpouringdiaphragm, basic mechanical model for steel can-tilever beam under two critical load cases has beenestablished and the analytical expression of the opti-mal action range of transverse prestressed tendons
has been deduced according to plane cross-sectionassumption.
(3) In this paper, an optimization design scheme based onthe stress at steel-concrete interface has been given,which is applied to a real bridge. The analysis resultsindicate that using this optimization design schemeto make optimization design on steel cantilever couldget the scheme that meets design requirements withreasonable stress.Thismethod could reduce unneces-sary trials and save computing resource to the greatestextent, so as to realize rapid and accurate design.
This structural optimization method provides the theo-retical support to the promotion of steel cantilever wideningconcrete box girder method and also promotes the applica-tion of structural optimization theory in bridge design.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Mathematical Problems in Engineering 13
Acknowledgments
The research described in this paper was financially sup-ported by the science and technology funds of LiaoningEducation Department (20131021) and the National NaturalScience Foundation of China (51308090).
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