an optimized design of network arch bridge using …...width, bw span, l rise, r h figure 1.3d view...

1. INTRODUCTIONNetwork arch bridges seem to be very competitive forroad bridges of spans of 135 m to 160 m, due to itsbeneficial structural behavior which is mainly subjectedto axial forces (Tveit 2003). Furthermore, the highstiffness and therefore small deflections favor theapplication of this kind of bridges for high speedrailway as well as roadway transportations. Research onoptimizing realistically three-dimensional structuresespecially large structures, is now feasible due toadvances in numerical optimization methods, computerbased numerical tools for analysis of structures andavailability of powerful computing hardware. Thus,

Advances in Structural Engineering Vol. 17 No. 2 2014 197

An Optimized Design of Network Arch Bridge using

Global Optimization Algorithm

Nazrul Islam1,*, Shohel Rana2, Raquib Ahsan3 and Sayeed Nurul Ghani4

1Department of Civil, Construction and Environmental Engineering, North Carolina State University, Raleigh, NC 27606, USA2Bridge and Structure Laboratory, Department of Civil Engineering, The University of Tokyo, Tokyo 113-8656, Japan

3Department of Civil Engineering, BUET, Dhaka-1000, Bangladesh4Sayeed NurulGhani, Tucson, Arizona, USA

(Received: 1 April 2012; Received revised form: 14 December 2013; Accepted: 14 December 2013)

Abstract: An optimization approach of network arch bridges using a globaloptimization algorithm EVOP is presented in this paper. The objective is to minimizethe cost of superstructure particularly arches and hangers of network arch bridge byoptimizing the geometric shape, rise to span ratio, cross section of arch, and thenumber, arrangement and cross sectional dimensions of hangers. Constraints fordesign are formulated as per AASHTO, AISC and ACI Specifications. The minimumcost design problem is characterized by having a combination of continuous, discrete,and integer sets of design variables and is subjected to highly nonlinear, implicit anddiscontinuous constraints. An optimization algorithm, evolutionary operation (EVOP),is used that is capable of locating directly with high probability the global minimumwithout requiring information on gradient or sub-gradient of the objective function.EVOP is interfaced with finite element analysis software, ANSYS for evaluation ofstructural response of the bridge to verify the constraints and also to evaluate objectivefunction. Within the range of design constant parameters considered, it is observedthat 38% to 40% of total cost can be saved for circular and parabolic archesrespectively, if design is optimized. Results also show that parabolic arch withoptimum design variables is more economic than the optimized arch bridges withcircular arch geometry.

Key words: global optimization, EVOP, arch bridges, structural optimization, finite element analysis.

large and important projects like network arch bridgeshave the potential for substantial cost reduction throughapplication of optimum design methodology.

A few researches regarding optimization of networkarch bridges have been performed in recent years. Tveit(1980) developed some concepts regarding hangerarrangement to reduce bending moments in arch andalso in hangers. Brunn et al. (2003) adopted ahypothesis for optimization of hanger arrangement thatarch should be a part of a circle and hanger should bearranged in such a way that hanger intersections lie onthe radii of the arch circle. They adopted trial and errorbased traditional method for optimization of various

*Corresponding author. Email address: [email protected]/[email protected]; Tel: 919-798-4091.

(2002), AASHTO (2002) and AISC (2005) designconsiderations. Design variables and design constantparameters are specifically shown in Figures 1–6.

3. OPTIMIZATION PROCESSNine numbers of design variables and seven numbers ofimplicit constraints are associated with the bridgeoptimization problem under discussion. The designvariables are a combination of continuous, discrete andinteger types. The minimum cost design problem issubjected to highly nonlinear, implicit anddiscontinuous constraint having multiple local minimawhich requires an optimization method to derive theglobal optimum. To deal with this optimizationproblem, the global optimization algorithm namedEVOP is used. It has the capability to locate directlywith high probability the global minimum. It has theability to minimize directly an objective functionwithout requiring information on gradient or sub-gradient. There is no requirement for scaling ofobjective and constraining functions. The method isgood to minimize functions of mixed continuous,discrete and integer variables. It is ideally suited forsystem optimization of real life design. It does notrequire training and initial known population size asrequired by genetic algorithm. Objective andconstraining functions can possess finite number ofdiscontinuities. It has facility for automatic restarts tocheck whether the previously obtained minimum is theglobal minimum. EVOP was extensively tested on morethan fifty test functions well known for their numericaldifficulties and resistance to locating the minimum.Detail and nature of these test functions are available inGhani (2008).

The problem formulation is as follows.

Find design variables, (5)

to minimize the objective function, F(x)

subjected to implicit constraints,(6)

with explicit constraints on design variables, (7)

where i = 1, 2… … NIC; k = 1, 2 … … NDIV; NIC istotal number of implicit constraints; NDIV is total

x x xkL

k kU≤ ≤

G x G x G xiL

i iU( ) ( ) ( )≤ ≤

x x x x xk NDIV= { }1 2 3, , ..........

198 Advances in Structural Engineering Vol. 17 No. 2 2014

An Optimized Design of Network Arch Bridge using Global Optimization Algorithm

design parameters. Again, Tveit (2008) suggested thatoptimized network arches can be achieved if somehangers cross other hangers at least twice.

Previous works on optimization of the network archbridges with inclined hangers has been adopted basedon some hypothesis and followed traditional method ofdesign. One of the main limitations of traditionaloptimization approaches is that they can easily beentrapped in local minima i.e. best possible solutionmay not be achieved. As a result, traditional approacheshave difficulties to determine the global optimum.Global optimization of network arch bridge systemconsidering all possible design variables has not yetbeen performed. In this study, simulation drivenoptimization of the present bridge system is performedby adopting an evolutionary based global optimizationmethod, EVOP developed by Ghani (1989) which hashigh probability for seeking best possible solution.

2. OPTIMAL DESIGN PROBLEMSTATEMENT

In this study, the objective of design is costminimization of superstructure particularly arch andhangers of network arch bridges by taking into accountcosts of materials, fabrication and installation. Unitcosts used in this study are based on Roads andHighway Department cost schedule (RHD 2008).Totalcost is determined as:

CT = CHC + CAC + CAS (1)

where, CHC, CAC and CAS are the costs of cable ofhangers, concrete section of arch and amount ofreinforcement required all over the arch respectively.Costs of individual components are calculated as:

CHC = UPHCWHC (2)

CAC = UPACVAC (3)

CAS = UPASWAS (4)

where, UPHC, UPAC and UPAS are the unit prices ofmaterials, fabrication and installation of hangers,concrete of arch and the reinforcement required in thearch respectively; WHC, VAC and WAS are the weight ofthe cables, volume of the concrete required in archesand weight of reinforcement required in archesrespectively.

Design variables with explicit constraints, designconstant parameters and implicit constraints for theoptimization problem are enlisted in Tables 1–3, whereupper and lower bounds of constraints come from ACI

number of independent design variables. G(x)i and F(x)are implicit constraints and objective functionrespectively. The functions G(x)i and F(x) have beenevaluated from finite element simulation in the presentstudy. The lower and upper bound of explicit constraints

and implicit constraints may be either constants orfunctions of independent variables. The implicitconstraints are allowed to make the feasible vector-space non-convex.

There are six fundamental processes in the EVOP,


Nazrul Islam, Shohel Rana, Raquib Ahsan and Sayeed Nurul Ghani

Table 1. Design variables with explicit constraints

Explicit Constraint______________________________

Design Variables Variable Type Lower bound Upper Bound

Number of Hangers (Each Arch), Nh Integer 4 60Start Angle for Hanger Inclination Set1, ϕ1 Continuous 0° 80°Start Angle for Hanger Inclination Set2, ϕ2 Continuous 0° 80°Angle Change for ϕ1, ∆ϕ1 Continuous –2° 2°Angle Change for ϕ2, ∆ϕ2 Continuous –2° 2°Cross Sectional Area of Cable of Hanger, Ah Discrete 96.8 mm2 2929 mm2

Arch Width, Bh Discrete 250 mm 3000 mmArch Depth, Hh Discrete 250 mm 4000 mmRise to Span Ratio, Rh Continuous 0.14 0.25

Table 2. Design constant parameters

Design Constant Parameters Value

Modulus of Elasticity for Concrete, EC 24800 MPaPoisson’s Ratio of Concrete, υ 0.2

Material Properties Concrete Compressive Strength at 28 days (Arch and Bracing), fca’ 25 MPa

Concrete Compressive Strength at 28 days (Deck), fcd’ 50 MPa

Ultimate Strength of Cable of Hanger, fu 1520 MPaArch Reinforcement Yield Stress, fy 413 MPa

Span, L 100 m

Geometric Properties Width of Bridge, Bw 10 mArch Section RectangularArch Shape Circular, Parabolic

Standard Vehicle Load AASHTO HS 20-44 Single and LaneLoading No of Lane Two

Wearing Surface 1436 N/m2

Unit Price for Cable of Hanger, UPHC 223 BDT/kgCost Parameters Unit Price for Concrete of Arch, UPAC 10527 BDT/m3

Unit Price for Reinforcement of Arch, UPAS 80 BDT/kg

General Design Code AASHTO (2002), AISC (2005)

Table 3. Implicit constraints

Implicit Constraint______________________________________

Response Lower bound Upper Bound

Extreme Hanger Stress, σmax 0 0.75 Fu or 0.9 FyStrength Criteria of Arch, CRT 0 1Design Reinforcement Factor in Arch, RNR 1% 8%End Angle for Hanger Inclination Set1, ϕ1n –80° 80°End Angle for Hanger Inclination Set2, ϕ2n –80° 80°In Plane Slenderness ratio 0.1 22Arch Depth/ Width Ratio 0.25 6

viz. Generation of a ‘complex’, Selection of a ‘complex’vertex for penalization, Testing for collapse of a‘complex’, Dealing with a collapsed ‘complex’,Movement of a ‘complex’ and Convergence tests. Detaildescription of the six processes can be found in Ghani

(1989). To understand the algorithm, some informationregarding “complex” is described here. A ‘complex’ isan object which occupies an N-dimensional parameterspace defined by K ≥ (N + 1) vertices inside a feasibleregion where, N is the number of design variables. It canrapidly change its shape and size for negotiatingdifficult terrain and has the intelligence to move towardsa minimum. Figure 7 (Rana et al. 2013) shows a‘complex’ with four vertices in a two dimensionalparameter space (X1 and X2 axes). X1 and X2 are the twoindependent design variables. The ‘complex’ verticesare identified by lower case letters ‘a’, ‘b’, ‘c’ and ‘d’ inan ascending order of function values, i.e. F(a) < F(b)< F(c) < F(d). Each of the vertices has two co-ordinates-(X1, X2). Straight line parallel to the co-ordinate axes areexplicit constraints with fixed upper and lower limits.The curved lines represent implicit constraints set toeither upper or lower limits. The hatched area is the twodimensional feasible region. The general outline of EVOPalgorithm is illustrated in Figure 8 (Ahsan et al. 2012).

At first, all (k – 1) vertices of the complex that satisfyall explicit and all implicit constraints are randomlygenerated beginning from a single feasible starting



Width, Bw Span, L

Rise, Rh

Figure 1. 3D view of network arch bridge

Span, L

Rise, Rh

N1

N4

N1 = Nh

Figure 2. Typical hanger arrangement

Start Angle, 1ϕ

ϕ'1 ϕ"1 ϕ'1 ϕ"1− = ϕ1∆

Figure 3. Hanger set 1 and its vertical inclination

Start angle, 2ϕ

ϕ'2 ϕ"2 ϕ'2 ϕ"2− = ϕ2∆

Figure 4. Hanger set 2 and its vertical inclination

Si

ϕ'iSi = Sj Sj

Figure 5. Alternate hanger position from one of equidistant hanger

nodes along the arch

Hanger

8000Two lane roadway

80009000

10000

10001000

120400

Arch section

Figure 6. Deck section (Dimension in mm)

ab

cd

G1 (x) = U1

G2 (x) = U2

G1 (x) = L1

G2 (x) = L2

X2

u2

u1 X1

l2

l1

Figure 7. A “complex” with four vertices

point. Next, the worst vertex of the ‘complex’ (thevertex with the highest function value) is penalized byover-reflecting on the centroid of the complex. For thispenalization to be successful, testing for collapse of the‘complex’ is performed. A ‘complex’ is said to havecollapsed in a subspace if the i th coordinate ofthe centroid is identical to the same of all ‘k’ vertices ofthe ‘complex’. This is a sufficiency condition anddetects collapse of a ‘complex’ when it lies parallelalong a coordinate axis. Once a ‘complex’ has collapsedin a subspace it will never again be able to span theoriginal N-dimensional space. After detecting thecollapse of a ‘complex’ on to a subspace some actionsare taken such that a new full sized ‘complex’ isgenerated. It now spans the n-dimensional feasiblespace as defined by the explicit and implicit constraints.The ‘complex’ is next moved. The movement ofcomplex process begins by over-reflecting the worstvertex of a ‘complex’ on the feasible centroid of theremaining vertices to generate a new trial point. If therefection step is unsuccessful, then some contractionsteps and expansion step are applied sequentially togenerate a new feasible trial point using the contractioncoefficient (β ) and expansion coefficient (γ )respectively. However, before penalizing a worst vertexthe feasibility of the centroid of the remaining (k–1)vertices is checked. If it is infeasible appropriate steps

are taken to regain feasibility. While executing theprocess of movement of a ‘complex’, tests forconvergence are made periodically after certain presetnumber of calls to the objective function.

To interface the finite element simulation withevolutionary operation, EVOP which is originallywritten in FORTRAN, a platform is established byvisual C++. The platform is used to transfer theparameters from EVOP to the simulator input file and toextract the response values of interest from thesimulator’s output file for return to EVOP. In the wholeoptimization process, the platform in visual C++, EVOPin FORTRAN and ANSYS (2009) are interlinked in theprocess as shown in Figure 9.

Figure 10 shows optimization flow chart denoting fileinput output operations. An initial feasible designdetermined from conventional design of the systemwhich satisfies all implicit and explicit constraints, isinvoked through the interfacing platform which runsEVOP and acts as platform for data structure definition.Design variables are intelligently allocated by EVOP.EVOP transfers the parameters to user’s simulationcode through the interfacing platform. Simulationoutput file provides necessary information for EVOPthrough platform and updated design variables aredynamically decided by the optimization platform fornew simulation. The process is repeated until all of the



NoNo

Yes

Testing for collapse of a 'complex', and dealing with a collapsed 'complex'

Convergence test

Movement of a 'complex'

Generation of a 'complex'

Selection of a 'complex' vertex for penalization

Penalized Vertex

An initial feasiblevertex and EVOP control parameters

Limit of function evaluations exceed?

Stop Optimum Solution

Yes

Converged?

Figure 8. General outline of EVOP algorithm

simulation code runs required by the iterative study havebeen completed. The solution is then said to beconverged and optimum results are recorded. Abenchmark problem of Prasad and Haftka (1979) inVerification Manual of ANSYS “Shape Optimization ofa Cantilever Beam” is solved to check the EVOPoptimization process. Result in (Islam 2010) drawssuccessful verification of interfacing of EVOP withfinite element software, ANSYS on benchmarkoptimization problems.

4. FINITE ELEMENT ANALYSIS OFNETWORK ARCH BRIDGES

The complete 3D finite element model of the bridge hasbeen introduced by assigning appropriate element types

and geometries, meshing, assigning proper boundary andload conditions. ANSYS is chosen for finite elementsimulation of the arch bridge. Beam, link and solidelements have been used for the simulation of arch,hanger and deck of the bridge respectively. Surfaceelement is incorporated on deck nodes for 3D structuralsurface effect. A load family is assigned for the highwayload by the vehicle type AASHTO standard HS 20–44truck and lane load. The complete finite element model ofthe bridge is shown in Figures 11 and 12.

It has been considered that arch and arch bracing areto be made of reinforced concrete, hanger is made ofzinc-coated steel structural strand of ASTM A586(1998) standard and deck is of reinforced concrete. Inthe finite element modeling deck is introduced for



Design

parametersF

eedb

ack

for

desi

gnva

riabl

es

Response

parameters

Sim

ulat

or’s

resp

onse

Objective

function

Implicit

constraintfunction

Explicit

constraintfunction

EV

OP

controlparam

eters

FE simulator ANSYS

Interfacing platformwritten in visual C++

EVOP written in FORTRAN

Figure 9. Black box interfacing

Initialdesignvariables

Updated

designvariables

Interfacing platform

EVOPdata processing

Simulationinput file

ANSYS user’s simulation code

Simulationoutput file

EVOPdata processing

Converged?No Yes

Optimumdesign

Obj

ectiv

e fu

nctio

n /

impl

icit

cons

trai

nts

Start initial feasible design

Figure 10. Optimization flow chart

vehicle movement and to transfer load to arch throughhanger. Material properties of reinforced concrete andcable are shown in Table 4. Vehicle movement andresponse of the bridge are shown in Figures 13–15.

After performing finite element analysis of networkarch bridges, ANSYS CivilFEM has been used forstructural evaluation and then the response is utilizedby the optimization algorithm for further dataprocessing. Scripts have been written to recordmaximum hanger stress of all available hangers for allpossible load combinations and considering all vehiclepositions acting on the deck. Similarly maximumpercentage of steel required in each arch element andtotal enveloped reinforcement, for all possible loadcombinations and considering all vehicle positionsacting on the deck, is recorded following design of archusing CivilFEM. From post processing consideration,two different scripts of FE analysis and design arewritten and results from analysis are post processedsubsequently which is available for object function and

implicit function of EVOP. A series of functionevaluation is required by the optimization algorithm,EVOP and EVOP guides finite element software foranalysis of the structure by batch execution andterminate after each data processing. Total iteration ofFE analysis is completed when the result ofoptimization is converged to its optimum design. Ahigh end machine has been used in this study forreducing computational time in the optimizationprocess of fine element model of network arch bridges.The convergence of the optimization algorithm wasfound fast. Two-three restart of the algorithm is foundenough for the convergence. In addition, in each restartof EVOP 100–500 FE analysis is carried out by thealgorithm depending on the initial starting point.

5. RESULTS OF OPTIMIZATIONOptimum design parameters for circular and parabolicarch bridges compared with initial design is shown inFigures 16 and 17 and in Table 5. Influence lines of



Four noded surface element on deck

Beam element for arch

Beam element forbracing

Link element for hanger

Eight noded solid element for deck

Figure 11. Typical finite element model of network arch bridges

0.000

7.500 22.500

15.000 30.000 (m)

Z

YX

Figure 12. Finite element mesh in deck



Table 4. Material properties

Material Parameters Value

Concrete Modulus of Elasticity for Concrete, Ec 24800 MPaPoisson’s Ratio, υ 0.2

Concrete Compressive Strength at 28 days (Arch and Bracing) 25 MPaConcrete Compressive Strength at 28 days (Deck) 50 MPa

Unit Weight 24.5 N/m3

Reinforcement Arch Reinforcement Yield Stress, fy 413 MPaModulus of Elasticity, Es 200 × 103 MPa

Poisson’s Ratio, υ 0.3Density, ρ 7833 kg/m3

Cable of Hanger Ultimate Strength of Cable of Hanger 1520 MPaModulus of Elasticity for Concrete, Ec 195 × 103 MPa

Poisson’s Ratio, υ 0.3Density, ρ 8000 kg/m3

Surface Elements Density, ρ 0.0 kg/m3

MN

MX

1.30 (DC + DW) + 2.17 (L + I)

−.208192−.185059

−.161927−.138794

−.115662−.092529

−.069397−.046264

−.023132.512E-06

Figure 13. Typical deflection for dead and live load

1

1.30 (DC + DW) + 2.17 (L + I)

−.808E + 08−.663E + 08

−.517E + 08−.371E + 08

−.226E + 08−.803E + 07

.654E + 07.211E + 08

.357E + 08.502E + 08

AUG 14 201001:08:19

NODAL SOLUTION

STEP = 39SUB = 1 TIME = 26SX (AVG) RSYS = SOLUDMX = .203453SMN = −.808E + 08 SMX= .502E + 08

Figure 14. Typical stress contour for dead and live load

bending moment for optimized arrangements andvertical hanger arrangements with the same archsection are compared in Figure 18 for circular archgeometry.

Ratio of influence line ordinate for bending atdifferent location is depicted in Figure 19 whereratio of influence line ordinate is defined as theinfluence line ordinate for vertical hanger



Figure 15. Typical multi load step for AASHTO HS 20-44 truck load

(a)

(b)

(c)

Angle vary from 39.62° to 36.97°

Angle vary from32.74° to 31.55°

Figure 16. Arch bridge with: (a) initial design variables; (b)

optimum design variables for circular arch; (c) optimum design

variables for parabolic arch

(a)

(b)

(c)

2800

600

Maximum2.48% steel(Circular arch)

Maximum1.74% steel(Parabolic arch)

Maximum2.41% steel

450

1600

Maximum2.82% steel

350

1975

Figure 17. Arch section: (a) before optimization; (b) after

optimization for circular arch; (c) after optimization for parabolic

arch (all dimensions are in mm)

arrangement divided by that ordinate for optimizedhanger arrangement; K as shown in that figure is afactor for position of arch along the span, L such thatfor any distance of arch position X, K = X/L. Thefigure shows that at different arch position influenceline ordinate (IL) for vertical hanger arrangement is

approximately 2 to 40 times larger than that ofoptimized hanger arrangement.

Bending moments and reinforcement percentagerequired in the arch for specific load step are shown inFigure 20 which concludes that response parameters inthe arch with optimized design variables are negligible



Table 5. Design variables for before and after optimization

After After Before optimization optimization

Parameter Unit optimization (Circular) (Parabolic)

Number of Hangers, Nh — 26 20 22Start Angle for Hanger Inclination Set1, ϕ1 Degree 0 39.62 32.74Start Angle for Hanger Inclination Set2, ϕ2 Degree 0 36.97 31.55

Angle Change for ϕ1, ∆ϕ1 Degree 0 –0.265 –0.11

Design Variables Angle Change for ϕ2, ∆ϕ2 Degree 0 0.265 0.11Cross Sectional Area of Cable of Hanger, Ah mm2 800 1548.4 1025.8

Arch Width, Bh mm 600 450 350Arch Depth, Ha mm 2800 1600 1975

Rise to Span Ratio, Rh — 0.18 0.2105 0.2386

0.6 0.8 1

0.4 0.6 0.8 1

2

1

0

−1

−2

−3

2

1

0

−1

−2

−3

Influ

ence

line

ord

inat

e of

ben

ding

mom

ent,

N-m

Optimal hangerarrangementVertical hangerarrangement

Optimal hangerarrangement

Vertical hangerarrangement

0.4

0.2

Xi = KLUnit load

xp = apLL

0.2

(a)

2

0.2 0.4 0.6 0.8 1

1

0

−1

−2

−3

Optimal hangerarrangement

Vertical hangerarrangement

(b)

(c)

(d)

Figure 18. (a) Location of influence line along the arch; (b) Influence line for bending moment at X = 0.2L; (c) Influence line for bending

moment at X = 0.3L; (d) Influence line for bending moment at X = 0.4L

than that of arch with vertical hangers. Reinforcementpercentage envelope required along the arch for allload steps is shown in Figure 21 which shows thatvertical hanger arrangement requires larger amount of

steel compared with optimized hanger arrangements.Results also show that maximum amount of steel isrequired at quarter span for vertical hangerarrangements.



50

IL for BM at K = 0.1IL for BM at K = 0.4IL for BM at K = 0.7



40

30

20

10

0

−10

−20

−30

−40

−500 0.2

Rat

io o

f inf

luen

ce li

ne o

rdin

ate

0.4ap

0.6 0.8 1

Figure 19. Ratio of influence line ordinate for bending moment at X = KL

Distance from left supportof Arch (m)

Distance from leftsupport of Arch (m)

Distance from leftsupport of Arch (m)

Ben

ding

mom

ent,

(N-m

)

Ben

ding

mom

ent,

(N-m

)

Rei

nfor

cem

ent (

%)

0 10

−1.E + 07

−5.E + 06

0.E + 00

5.E + 06

1.E + 07

−1.E + 07

−5.E + 06

0.E + 00

5.E + 06

1.E + 07

10

8

6

4

2

00 10 20 30 40 50

20 30 40 50

Vertical hanger arrangementOptimal hanger arrangement

0 10 20 30 40 50

Optimal hanger arrangementVertical hanger arrangement

Optimal hanger arrangementVertical hanger arrangement

(a) (b)

(c)

Figure 20. (a) Bending moment diagram of arch for dead load only; (b) Bending moment diagram of arch for a load combination of dead

and live load; (c) Reinforcement percentage envelope of arch for a load combination of dead and live load

A cost breakdown for hanger, concrete of arch andarch reinforcement is outlined in Table 6 for initial andfinal optimized design which shows that 40.38% costcan be saved for circular arch and 38.1% cost can besaved for parabolic arch if design is optimized.

6. CONCLUSIONSBased on the numerical analysis within the range ofparameters examined, the following conclusions can bedrawn for structural optimization of network arch bridges:

i) It is found that optimized hanger arrangement isattained for hangers placed at some certaininclinations with vertical. Hanger inclination isfound same to all hangers for parabolic archgeometry whereas for circular arch geometry theinclination varies from one hanger to another ina small range. Under the scope of study, hangerinclination for parabolic arch geometry is foundto be 32 degree whereas it varies from 36 to39 degree from first hanger to the last forcircular arch geometry.

ii) Parabolic arch geometry is found to be moreeconomical than circular arch geometryconsidering optimum design.

iii) Under the scope of study, it is observed thatoptimized network arch bridge with circulararch geometry requires lesser number of hangersthan parabolic arch geometry.

iv) Under the scope of study, it is found thatoptimized network arch bridge with circulararch geometry requires shallower arch than thatrequired for parabolic arch geometry.

v) Within the range of design constant parametersit is observed that 36% to 40% of total cost canbe saved for circular and parabolic arches ifdesign is optimized.

Findings outlined here are limited for an arch bridgeof particular span and lane. Therefore it is recommendedthat the study can be extended further in the followingfields:

i) Optimization can be performed for differentspan and lane of bridge to have generalconclusion on global optimum design of archbridges.

ii) Dynamic properties of the bridge with optimumdesign variables may be investigated further.

iii) The study can be extended for PerformanceBased Optimality criteria as in Liang (2002).



15

10

55

01 5 10 14

Optimum designOptimal arch section with vertical hanger arrangement

19 23Distance from left support of Arch (m)

Rei

nfor

cem

ent p

erce

ntag

e

28 32 37 41 46 50

15

10

58

56

4

2

01 5 10 14 19 23

Distance from left support of Arch (m)

Rei

nfor

cem

ent p

erce

ntag

e

28 32 37 41 46 50

(a) Optimum designOptimal arch section with vertical hanger arrangement

(b)

Figure 21. Reinforcement percentage envelope along the arch of: (a) circular arch geometry; (b) parabolic arch geometry

Table 6. Result comparison for cost of initial and optimized design

Circular Arch Parabolic Arch

Cost Breakdown Initial Final Initial Final

Cost of Cable (BDT) 0.100001412E+07 0.199471517E+07 0.908678753E+06 0.154834904E+07Cost of Concrete of Arch (BDT) 0.417691526E+07 0.180762830E+07 0.409068897E+07 0.176910130E+07Cost of Reinforcement in Arch (BDT) 0.345129619E+07 0.134102902E+07 0.269647296E+07 0.145578683E+07

Total Cost (BDT) 0.862822557E+07 0.514337249E+07 0.769584069E+07 0.47650856E+07

40.38% Save of 38.1% Save of Initial Design Initial Design

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NOTATIONAh cross sectional area of cable of

hangera, b, c, d complex verticesap, xp variable for defining unit load to

determine influence line in archBh arch widthBw width of bridgeC location of centroid of ‘complex’CAC cost of concreteCAS cost of reinforcementCHC cost of cable of hangersCT total costCRT strength criteria of archDL dead loadEC modulus of elasticity for concreteEs modulus of elasticity for steelfca

’ concrete compressive strength at 28days for arch and bracing

fcd’ concrete compressive strength at 28

days for deckfu ultimate strength of cable of hangerfy arch reinforcement yield stressFy specified minimum yield stress of

the type of steel being usedf(x) objective functionf(a), f(b), f(c), f(d) function values of complex

vertices



G(x)i implicit valuesG(x)L

i upper limit of implicit constraintsHh arch depthk number of vertices in complex of

evopXi, Ki variable for defining position of

archL span of archLi, Ui lower limit and upper limit symbol

for constraintsNh total number of hangersNh1 number of hangers for set1 cable in

each archNDIV number of design variablesNIC number of implicit constraintsRh rise to span ratioRNR reinforcement factor of the archS length of element of archUPHC unit price of cableUPAC unit price of concrete of archUPAS unit price of steel of archVAC volume of concreteWAS weight of reinforcementWHC weight of cablesX the centroid of complex which lies

in the infeasible area

xk design variablesxL

k lower limit of design variablesxU

k upper limit of design variablesσmax extreme hanger stressϕ1 start angle for set 1 cableϕ2 start angle for set 2 cable∆ϕ1 angle change for set 1 cable∆ϕ2 angle change for set 2 cableϕΤ1n end angle for hanger inclination

set1ϕΤ2n end angle for hanger inclination

set2υ Poisson’s ratio of concreteρ densityAASHTO American association of state

highway and transportation officialsAISC American institute of steel

constructionAPDL ANSYS parametric design

languageASTM American society for testing and

materialsEVOP evolutionary operationFEA finite element analysisLRFD load and resistance factor designRHD roads and highway department



an optimized design of network arch bridge using …...width, bw span, l rise, r h figure 1.3d view...

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