Computers and Structures 170 (2016) 13–25

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Computers and Structures

journal homepage: www.elsevier .com/locate /compstruc

An optimization model for the design of network arch bridges

http://dx.doi.org/10.1016/j.compstruc.2016.03.0110045-7949/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: d.bruno@unical.it (D. Bruno), paolo.lonetti@unical.it

(P. Lonetti), arturo.pascuzzo@unical.it (A. Pascuzzo).

D. Bruno, P. Lonetti ⇑, A. PascuzzoDepartment of Civil Engineering, University of Calabria, Via P. Bucci, Cubo39-B, 87030 Rende, Cosenza, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 August 2015Accepted 28 March 2016Available online 16 April 2016

Keywords:Network arch bridgesStructural optimizationFinite element analysisDesignSizing optimization

A new design methodology, which evaluates the optimum configuration of network arch bridge schemesis proposed. A three-step optimization algorithm is implemented in a FE model, with the purpose toevaluate the bridge optimum configuration, involving the lowest material quantity and the best strengthperformance level in all structural members of the bridge. The stability and the efficiency of the formu-lation were verified with respect to several bridge configurations ranging from small to large spans.Moreover, parametric results are presented to investigate the interaction between cable-system, girderand arch, giving rise to specific analyses useful for design purposes.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Tied arch bridges can be considered as enhanced constructionschemes, which are able to provide aesthetic, structural and eco-nomic performances to overcome small, medium and large spans[1]. Most of the existing bridge configurations consist of an archand a girder, whose internal transferring forces are guaranteedby the cable system, typically formed by cable elements. The cablearrangement plays a fundamental role in the structural behavior,since it is able to strongly influence the internal stress distributionas well as the deformability properties of the entire bridge struc-ture [2]. Several hanger geometries, such as vertical, inclinedV-shaped or network, are frequently utilized for design purposesin tied arch bridges. From the structural point of view, V-shapedor network (inclined hangers with multiple intersections) archbridges are preferred to conventional vertical hanger bridgetypologies, since they are able to guarantee a high efficientresponse, which minimizes bending effects in both arch and girder[3]. However, the hanger arrangement, especially in network archbridges, can be considered as a complex structural system, whoseelements, i.e. the hangers, interact by means of tension only inter-nal forces with girder and arch [4]. In particular, each element ofthe cable-system is affected by geometrical nonlinearities arisingfrom cable sag effects, which strongly influence the actual stressdistribution in the bridge components. As a consequence, a funda-mental task to be achieved is the evaluation of the initial configu-ration under dead loads in terms of internal stresses and strains of

bridge constituents, by means of a nonlinear field model, avoidingunexpected and unrealistic stresses distribution in the cable due tocompressive forces. Moreover, it is required to identify the opti-mum design configuration consistently with a performance basedapproach, in which a proper choice of hangers, arch and girderdimensioning should be determined to take into account externalloads [5,6].

In the literature, most of the analyses are carried out for con-ventional vertical hanger bridge schemes, in which heuristic mod-els and preliminary design rules can be adopted because of thequite standard stress distribution in the bridge components. As amatter of fact, under live loads, the design of the main constituentsis not complex at all, since the cable elements interact with thearch and girder by means of uncoupled vertical forces. In thisframework, the evaluation of the initial configuration can be iden-tified by using traditional ‘‘zero displacement” method, enforcingthe girder to remain straight under the application of the deadloads [7–11]. The extension of the structural analysis in the frame-work of arch bridges with inclined hanger elements is not straight-forward, especially in those cases in which a large number ofvariables is involved in the analysis, i.e. in long span bridges. In thisframework, the determination of the initial cable configurationunder dead loads as well as the design of the bridge componentsunder the action of the external loads should be considered as animportant task to be achieved. A review of the literature dealingwith the analysis of network arch bridges denotes that althoughsuch structures are receiving much attention in the last decades,many points still remain to be addressed satisfactorily. Most ofthe present studies on network arch bridges propose designspecifications and guidelines on bridge dimensioning by means

14 D. Bruno et al. / Computers and Structures 170 (2016) 13–25

of preliminary design rules [3,4]. Notable parametric studies andrelevant guidelines useful for design purposes are proposed in[2], in which investigations, in terms of arrangement of the hang-ers, arch configuration and geometrical characteristics of thebridge constituents, are carried out. Moreover, fatigue behaviorof cable system elements in terms of hanger arrangements is inves-tigated in [12], in which comparisons, in terms of hanger distribu-tion based on radial, constant or constant change slopeconfigurations are proposed. However, most of the models avail-able in the literature do not enter in detail in the calculation ofthe bridge configuration under dead loads, in terms of both initialcable force distribution and arch-tie geometric profile. The identi-fication of such configuration is quite important, in relationship tothe nonlinear behavior of the cable-system elements, which couldbe affected, under the external loads, by unexpected relaxationeffects of the hangers, producing overstressing in the adjoininghanger elements and in both girder and arch [5,13]. Anotherimportant issue is that preliminary design rules, available fromthe literature, provide information on a reasonable dimensioningof the structural components, but not on the best distribution ofmaterial according to the Performance Based Approach (PBA) [14].

Currently, tied arch bridges are designed by using conventionalmethodologies, which consist of heuristic procedures based on theexperience and expertise of the designer. Although parametricstudies are carried out on several classes of structures, the proce-dure to reach the best possible performance design solution stillremains extensive and quite difficult to be achieved. In the frame-work of tied arch bridges, to the best Authors’ knowledge only veryfew models are concerned to investigate the optimum bridge con-figuration. Preliminary works are developed in [2,15], in whichparametric studies in terms of hanger arrangement with the pur-pose to minimize bending moments in both arch and girder arepresented. Moreover, a design methodology based on classicalstructural optimization is proposed in [16], in which the optimumconfiguration of vertical hanger bridge arrangement is discussed interms of hanger cable-force distribution and material quantityinvolved in the bridge constituents. The optimum design configu-ration of network arch bridges is investigated in [17], in whichan optimization method is developed to minimize the cost ofsuperstructure (arch and hangers) in terms of geometric shape, riseto span ratio, cross sections of arch and hangers. In this framework,relevant results and guidelines for the design of network archbridges are proposed. Optimization methods are typicallyemployed to determine the bridge dimensioning, by minimizinga convex scalar function, which combines variables concerningthe geometry of the structure and the internal stress/strain distri-bution. However, the use of pure optimization methods, especiallyin the case of complex and large structures, such as the networkarch bridges, are affected by convergence problems in the solvingprocedure, due to the large number of variables. Moreover, theconsistency of the solution is not guaranteed in the final optimumconfiguration, since the solving procedure may lead to a local min-imum or unpractical results from the engineering point of view[10,14,18]. Alternatively, advanced formulations based on meta-heuristic algorithms are frequently utilized in those cases in which,multivariable or multicriteria affect the final optimum configura-tion. However, although the basic idea of heuristic methods isconceptually simple, the application of a generic metaheuristic toa generalized optimization problem requires a laborious imple-mentation with specific guidelines, which introduce numericalcomplexities in the formulation. Therefore, in order to avoid suchproblems, in the present paper a design procedure based on aniterative methodology developed in the framework ofperformance-based optimization techniques is proposed. Despiteexisting methods available from the literature, a simple and effec-tive methodology, easy to be implemented in several FE software,

is proposed. The design variables are the post tensioning forces inthe hangers and the initial strains in the girder and arch, whichidentify to the initial configuration under dead and permanentloads. Moreover, under the external loads, bridge geometric char-acteristics, which involve the lowest possible material in the cablesystem, girder, and arch and verify prescriptions arising fromexternal loads, are determined. In order to prove the effectivenessof the proposed model, parametric studies in terms of cable systemconfigurations on several bridge schemes are proposed. The outlineof the paper is as follows. In Section 2, the formulation of thedesign methodology, bridge modeling, together with the descrip-tion of the iterative procedure is presented. In Section 3, numericaldetails on the design method are reported, whereas in Section 4,numerical comparisons and parametric results are presented.

2. Formulation of the procedure

2.1. Bridge modeling

The bridge typology, reported in Fig. 1a, refers to a generalizedtied arch scheme, in which arch and girder are connected betweenthemselves at the bridge extremities, whereas the girder isassumed to be simply supported to the foundation system. More-over, the hangers may be arranged in various configurations suchas vertical, V-shaped or network. Without loss of generality, thearch and girder are assumed to be in steel, whereas hangers aremade of steel cable, requiring prestressing forces. The proposedoptimization method, is presented for a Network Arch Bridge(NAB) scheme, which is, in comparison to the existing ones basedon vertical or V-shaped (Fig. 1b), the most complex configurationto be analyzed. However, the theoretical formulation of the pro-posed model is quite general to be implemented also for conven-tional vertical or V-shaped cable arrangements.

The proposed strategy, despite existing optimization methods,identifies the optimum configuration on the basis of a step-by-step procedure, in which the solution is enforced by using physi-cally based expressions. The heuristic nature of the proposedprocedure does not ensure a priori that the predicted solution isa global optima. However, the optimum solution is determined,iteratively, by means of successive approximations of the final con-figuration by solving separate optimization problems implementedin different substeps. The proposed design methodology is basedon a three-step analysis, described in the next three subpara-graphs, in which the optimum solution is determined by usingan iterative procedure based on results obtained under Dead Loads(DL) and Live Loads (LL) combinations. In the present approach thedesign variables are the initial stresses in the hangers under DL, thecross sections of the hangers, girder and arch, whereas the charac-teristics of the cable-system (angle, number of cables) are typicallyassumed by the designer due to aesthetic requirements and thusare not included as variables in the optimization procedure.

2.2. Analysis under the action of dead loads (STEP 1)

Under the action of DL, it is required to evaluate internal stres-ses and deformations, which enforce the prescribed design geom-etry, known in the literature as ‘‘zero configuration”. In thisframework, the unknown quantities are represented by the inter-nal stresses of the cable-system elements, the initial position ofthe arch and the girder, which are determined in such a way toreproduce the design undeformed configuration. The hangersshould be designed in terms of post-tensioning forces to reproducethe initial configuration under dead loads, i.e. zero configuration.Moreover, the cross-sections are calculated to verify strengthprescriptions (maximum and fatigue stresses) under the action of

h

A

Δ G

H

X2

X1

b

(b)

(a)

Fig. 1. Bridge configuration and representative geometric lengths (a) existing configurations of tied arch bridges (b).

Fig. 2. Design variables involved in the optimization problem to determine theinitial configuration under DL.

D. Bruno et al. / Computers and Structures 170 (2016) 13–25 15

the live loads. The use of an optimization algorithm to simultane-ously design hangers under both dead and live loads may lead tosevere computational efforts with convergence problems in thesolving procedure. A design procedure based on different stepsseems to circumvent the numerical difficulties to achieve the finalsolution. Moreover, the division in two substeps is quite consistentto the current design approach in bridge engineering, in which atfirst, the zero configuration and subsequently the behavior underthe action of live loads are evaluated. Finally, improvements onthe convergence behavior, obtained by using the iterative proce-dure, can be explained in relationship to the dominant truss behav-ior of the cable system, which partially reduces the couplingbehavior between each element of the cable system. It is worthnoting that the number of unknown quantities, represented bypost tensioning stresses in the cables, is larger than the numberof available constraint equations defined by enforcing zero verticaldisplacements at the intersection points between cable and girder.As a consequence, in order to determine the initial configuration, anumerical procedure for solving an indeterminate system of equa-tions is required. To this end, an optimization problem is imple-mented, in which the objective scalar valued function ðgÞ isdefined as the norm of girder vertical displacements under theaction of dead and permanent loads and the control variables arethe post tensioning forces in the hangers ðS

�CÞ. Moreover, constraint

equations are introduced for the final stresses of the hangers andthe deformations of the arch and girder. Therefore, the optimiza-tion problem can be expressed as follows:

minS�C g ¼ kUG

2ðX1Þk1; with ½S�C �T ¼ fSL1; . . . ; SLNL ; SR1; . . . ; S

RNRg

s:t: 0 6 SLi 6 SA;0 6 SRj 6 SA i ¼ 1 . . .NL; j ¼ 1 . . .NR

NG½EG0 ;

�UG1 � ¼ 0; NA½EA

0 ; hUA2 i� ¼ 0

8>>><>>>:

ð1Þ

where k � kp is the Lp norm of the function ð�Þ with p = 1, i.e.,

k � kp ¼ ðR LG

0 j � jpdX1Þ1=p

, h�i is the notation utilized to indicate the

average value of the function ð�Þ, i.e. h�i ¼ 1LA

R LA

0 ð�ÞdS, with LG andLA are the total length of the girder and the arch, X1 and S are thespatial coordinates fixed on the girder and arch profiles, respec-tively, UG

2ðX1Þ is the function of girder vertical displacements,

ðSRi ; SLj Þ and ðSRi ; SLj Þ are final and initial stresses of the generic hangerelement of the cable system measured from the Right (R) or Left (L)of the bridge extremities (Fig. 2), SA is the upper bound of the allow-able stresses involved in the hangers and ðNR;NLÞ are the corre-sponding total number of hangers involved in the cable systemfor the right (R) and left (L) orientations. Moreover, ðN

�G;N

�AÞ are

the constraint operators, which control the deformations in the arch

and girder under DL. In particular, in order to avoid arch shorteningand tie lengthening, proper values of initial axial strains in the archðEG

0 Þ and in the girder ðEA0 Þ are enforced to have zero horizontal

½UG1 ¼ UG

1ðX1 ¼ LÞ� and average vertical ðhUA2 iÞ displacements, in the

girder and the arch, respectively. A synoptic representation of thevariables involved in Eq. (1) and the corresponding notation isreported in Fig. 2.

2.3. Analysis under the action of live loads (STEP 2)

The design of structural components is developed on the basisof common prescriptions arising from Ultimate, Serviceabilityand Fatigue Limit State method, i.e. ULS, SLS, FLS, respectively.The main aim of this step is to evaluate the optimum solution interms of the lowest possible material quantity involved in thebridge components, taking into account both strength anddeformability prescriptions available from common codes onbridge structures [19–21]. In particular, the design methodologyshould be able to verify, for each hanger, the following conditionsarising from the limit state design:

max SLðRÞiðjÞ

h iULS

6 SA;max DSLðRÞiðjÞ

h iFLS

6 DSA; max jUG2 j

h iSLS

6 dGA ; ð2Þ

where i = 1, . . . ,NL or j = 1, . . . ,NR are used to identify Left (L) or Right(R) oriented hangers, respectively, SA and DSA are the maximum(strength) or incremental (fatigue) allowable stress values for eachcable and dGA is the maximum displacement (vertical) of the girder.

16 D. Bruno et al. / Computers and Structures 170 (2016) 13–25

The evaluation of the hanger cross-section vectorA�¼ fAR

1; . . . ;ARNR ;AL

1; . . . ;ALNLg, is developed, as described in Fig. 3,

by introducing two design factors for each element of the cable sys-

tem, namely ULðRÞiðjÞ and XLðRÞ

iðjÞ . In particular, ULðRÞiðjÞ modifies the cross

section of the generic hanger with the aim to verify recommenda-tions on the admissible strength, namely Eq. (2).1 and Eq. (2).2. In

addition, the factors XLðRÞiðjÞ take into account deformability prescrip-

tions on the girder, i.e. Eq. (2).3, increasing at the intersection pointswith the girder, the stiffness of those cables affected by displace-ments larger than the allowable ones. The evaluation of the vectorA�is achieved by using a secant approach, in which the current value

of the generic cross section is obtained as a function of the previousestimate, at the k � 1-th iteration, by means of the following multi-plicative relationship:

ðALðRÞÞkiðjÞ ¼ ½ULðRÞiðjÞ X

LðRÞiðjÞ �

kðALðRÞiðjÞ Þ

k�1; ði ¼ 1 . . .NL; j ¼ 1 . . .NRÞ ð3Þ

The factors ULðRÞiðjÞ are defined by means of a linear extrapolation

between the maximum applied stress values reached under LL andthe corresponding allowable strength. In order to verify prescrip-tions concerning both ULS and FLS, the worst design criterion isconsidered in the definition of the generic performance factor, asfollows:

½ULðRÞiðjÞ �

k¼maxmaxULS

ðSLðRÞLL ÞkiðjÞSA

;maxFLS

ðDSLðRÞLL ÞkiðjÞDSA

264

375;ði¼1 . . .NL;j¼1 . . .NRÞ

ð4Þwhere SLL and DSLL are the values of the stresses for the right (R) orthe left (L) oriented cables observed in the ULS or FLS combinations,

respectively. The prediction of the factors XLðRÞiðjÞ is strictly connected

to the following limit function, which indicates, at the genericcable/girder intersection point, the ratio between the current valueof girder deflection and the allowable value:

½ðgGLðRÞðX1ÞÞ�kiðjÞ ¼

maxSLS

ðjUG2iðjÞjÞ

k

LðRÞ

dGA� 1; ði ¼ 1 . . .NL; j ¼ 1 . . .NRÞ ð5Þ

At a generic point of the girder, prescriptions on bridgedeformability are satisfied if the functions gG

R or gGL are strictly neg-

ative; for these values, no increments of the cable system stiffness

are necessary and thus the factors XLðRÞiðjÞ are supposed to be equal to

Fig. 3. Identification of the performance factors and the design criteria of arch andgirder.

one. Contrarily, in the case of a positive value, an increment of theglobal bridge stiffness is required, which is performed by improv-ing the stiffness of the cable elements. Such stiffness incrementwill interact with the ones predicted in the arch and girder, as

described subsequently. In particular, the definition of the XLðRÞiðjÞ

for the generic element of the cable system is based on the follow-ing piecewise functions:

½XLðRÞiðjÞ �

k ¼1 if ½ðgG

LðRÞðX1ÞÞ�kiðjÞ 6 0

maxSLS

ðjUG2iðjÞ jÞ

k

LðRÞ

dGAif ½ðgG

LðRÞðX1ÞÞ�kiðjÞ > 0

8>><>>:

ð6Þ

Therefore,during the iterations,ULðRÞiðjÞ andXLðRÞ

iðjÞ areevaluated taking

into account several cases, reported concisely in Table 1, dependingif ULS/FLS or SLS prescriptions are satisfied or not. However, thefourth case excludes the over use of the material, since when bothstrengthanddeformability requirementsareverifieda lowerpredic-tion of the cross-section is necessarily enforced depending from theratio between current stress and allowable quantity. Such condition

is applied until the values ofULðRÞiðjÞ differ from the unity. The design of

both girder and arch is considered by solving two uncoupled opti-mization problems, in which the lowest material quantity involvedin such bridge components is achieved. In particular, the objectivescalar valued functions, which are minimized during the solvingprocedure, correspond to the required steel quantities involved in

thebridge components, i.e.QG orQ A.Moreover, the control variablesare represented by the characteristic lengths, which describe thecross section shape, i.e. ðtG1 ; tG2 ; . . . ; tGnG Þ and ðt A1 ; t A2 ; . . . ; t AnA Þ. Finally,constrain equations, concerning the design criteria adopted for thearch or the girder, i.e. DA or DG complete the optimization problem,which is described by the following expressions:

minAGðAÞ

QGðAÞ ¼ kcGðAÞAGðAÞk1s:t: AGðtG1 ; tG2 ; . . . ; tGnG Þ > 0;AAðt A1 ; t A2 ; . . . ; t AnA Þ > 0;

DAj 6 aA;D

Gj 6 aG; j ¼ 1 . . .NULS

8>>><>>>:

ð7Þ

with

ðaGÞk ¼ ðaGÞk�1min 1;

dG2maxX12G

½jUG2ðX1Þj�

0B@

1CA; ðaAÞk

¼ ðaAÞk�11;

dA2

maxX12A

½jUA2 ðX1Þj�

0B@

1CA ð8Þ

where k � kp is the Lp norm of the function ð�Þ with p = 1, cG;A is the

weight density of the girder or the arch AG and AA are the corre-sponding cross section areas, DA

j and DGj represent the j-th values

of the design criterion involved in the bridge components andNULS is the number of loading combinations defined by the designrecommendations. Moreover, aA and aG correspond to the limitdesign thresholds, which modify the allowable maximum stresslevels in the bridge components on the basis of the requirementachievements on bridge deformability. In particular, aA and aG areequal to one, when bridge displacements are in agreement withthe serviceability displacement prescriptions; in such cases, thedesign of the bridge components, i.e. arch and girder, are developedessentially with respect to the maximum strength criterion.Contrarily, when deformability prescriptions are not verified, anincrement of current stiffness is required and thus the design is per-formed consistently with the maximum displacement approach.Such task is performed by reducing the maximum stress level,

Table 1Synoptic representation of the procedure utilized to predict the factor ULðRÞ

iðjÞ and XLðRÞiðjÞ in the STEP 2.

Case 1 2 3 4

Verified Predicted Verified Predicted Predicted Predicted Verified Predicted

(Strength) ULS/FLS NO ULðRÞiðjÞ > 1 NO ULðRÞ

iðjÞ > 1 YES ULðRÞiðjÞ = 1 YES ULðRÞ

iðjÞ 6 1

(Deformability) SLS NO XLðRÞiðjÞ > 1 YES XLðRÞ

iðjÞ = 1 NO XLðRÞiðjÞ > 1 YES XLðRÞ

iðjÞ = 1

D. Bruno et al. / Computers and Structures 170 (2016) 13–25 17

reached in the bridge, below the value provided by the design stresscriterion. It is worth noting that Eqs. (6) and (8) enforce the designconditions on strength or deformability on the basis of a secantapproximation, whose final optimum configuration is guaranteedby the iterative nature of the proposed algorithm. Such assumptionappears to be quite reasonable in relationship to the global behaviorof the network arch bridges, which typically are considered asprevalent truss structures with reduced bending moments [4] orat least a continuous beam with multiple elastic supports withrelative stiffness proportional to the mechanical characteristics ofthe hangers [10]. The connection between each hangers is dealtwith by means of a ‘‘needle-eye” device, which allows for hangerpassing through another. As a consequence the cable system isbased on a dominant truss behavior, in which a modification ofthe geometry in the hangers, produce small perturbations in thestress internal resultants of the adjoining ones.

2.4. Tolerance condition (STEP 3)

The tolerance conditions should be checked to verify if the cur-rent solution has reached the converged configuration. The presentstage should be considered after the evaluation of the initial con-figuration (STEP1) and the design of the bridge components underthe external loads (STEP 2). In particular, it is required to verify ifthe design obtained at the current iteration (k), in terms of cross-sections of the bridge components, does not differ from the oneobtained in the previous iteration step (k � 1) by means of thefollowing tolerance condition:

maxX

j¼1; . . . ;NR

i¼1; . . . ;NL ;

ðALðRÞiðjÞ Þ

k�ðALðRÞiðjÞ Þ

k�1

ðALðRÞiðjÞ Þ

k�1

24

35;ðA

GÞk�ðAGÞk�1

ðAGÞk�1 ;ðAAÞk�ðAAÞk�1

ðAAÞk�1

8>>>>><>>>>>:

9>>>>>=>>>>>;

6 toll:

ð9ÞIf Eq. (9) is not satisfied, an update of the k-th current solution, in

terms of bridge geometry and initial stresses, should be carried outand thus the algorithm continues to iterate. Contrarily, if toleranceconditions are satisfied, the final solution is achieved. A synopticrepresentation of the optimization procedure and the steps per-formed by the iterative procedure are reported in Fig. 4. Previousprocedure, presented for network systems, can be easily specializedfor other hanger arrangements, such as vertical or V-shaped archbridge schemes, because of the low complexities involved in thesecable systems. In particular, it is required to solve Eqs. (1) and (3) forthe evaluation of the initial configuration and cable dimensioning,respectively, taking into account of the correct numbering of theelements of the cable system. Moreover, governing equations con-cerning the optimization problems for the arch and girder, i.e. Eqs.(7), should be solved separately, since they are uncoupled with theanalysis developed on the cable system element.

3. Numerical implementation of the model

The proposed formulation, presented in previous section fromtheoretical point of view, is now described in its numerical imple-

mentation. In particular, the procedure consists of different steps,which are executed iteratively:

(1) generation of the finite element formulation and use ofpreliminary design rules;

(2) analysis under DL and identification of the initial configura-tion (STEP 1);

(3) calculation of the maximum stresses and displacementsunder LL and prediction of the new geometry of the struc-tural elements, i.e. arch, hangers and girder (STEP 2);

(4) check tolerance conditions and verify the consistency of thedesign configuration (STEP 3).

The steps are reproduced by an external subroutine, whichinteracts with Comsol Multiphysics and Matlab package [22,23].In particular, the algorithm was implemented by means of propercustomized script files, which manage the parameters and theresults required by the iterative procedure. The proposed formula-tion is quite general to be implemented in several computationalframeworks, since it is based on data and results, which can beeasily extracted and managed from many standard commercialFE software or by using conventional mathematical tools. In theproposed analyses, the numerical formulation of the bridge isbased on a FE model, in which the arch and the girder are basedon Timoshenko beam formulation, whereas the hangers are dis-cretized by using multiple truss elements in agreement with aMulti Element Cable System formulation (MECS) [24]. The stiffnessreduction caused by sagging is accounted by using Green Lagrangestrain measure by expressing the global strains in tangentialderivatives and projecting the global strains on the cable edge.Additional details on the approach here adopted to model nonlin-ear behavior of the cable elements can be found in [23]. The use of1-D model, with respect to more enhanced analyses based on 2D or3D FE formulations, appears to be more consistent with the pur-pose of the present study, mainly devoted to propose a simple con-sistent procedure from an engineering point of view. The initialdata, concerning the geometry and the mechanical properties ofthe FE model are assumed by using preliminary design rules, whichconsist of analytical expressions, obtained under very simplifiedassumptions and structural schemes [15]. It is worth noting thatthe use of such relationships is not mandatory, but it provides areasonable dimensioning for each structural element of the bridge,reducing the number of iterations to evaluate the optimum solu-tion. Once the FE model is generated, it is required to solve theoptimization procedure defined in Eq. (1), which corresponds tothe following nonlinear constrained optimization problem:

OBJ1 ! Min GðEG0 ; E

A0 ; S�

R; S�LÞ;

CE ! 0 6 SLi 6 SA;0 6 SRj 6 SA i ¼ 1 . . .NL; j ¼ 1 . . .NR

EQ ! þK�U�þS

�L�R þ S

�G0 þ S

�A0 ¼ F

�DL;

CE; NA ¼ 0 ! NA0 ðSRi ; SLj ;U�ÞE

A0 ¼ 0;

CE; NG ¼ 0 ! NG0ðSRi ; SLj ;U�ÞE

G0 ¼ 0:

ð10Þ

where G is the global objective function concerning the norm of ver-tical girder displacements as defined by Eq. (1), K

�is the stiffness

Fig. 4. Flow chart of design procedure.

18 D. Bruno et al. / Computers and Structures 170 (2016) 13–25

matrix, U�is the displacement vector, S

�G0 , S�

A0 and S

�L�R are the equiv-

alent stress resultant vectors produced by the initial strain distribu-tion in the girder, arch and cable system, respectively, and ðNA

0 ;NG0 Þ

are the constrain operator matrixes, which verify the initial bridgeconfiguration. Moreover, upper and lower bound constraints wereimposed on the design variables to ensure the accuracy of the expli-cit approximation. Since the proposed model is developed in theframework of a NL formulation, the optimization problem requiresan iterative approach to determine the current solution. The gov-erning equations concerning the STEP1 are solved in the frameworkof gradient algorithms based on SNOPT method, in which the opti-mum solution is computed by the evaluation of the gradients ofboth objective function and constraints by using numerical differ-entiation [23]. In particular, SNOPT uses a Sequential Quadratic Pro-gramming (SQP) algorithm, in which the objective function isassumed to be a quadratic polynomial, whereas the constraintsare treated as linear. Finally, the optimum solution is derived itera-tively by using the conjugate-gradient QP solver [25]. The analysisunder the action of LL was developed taking into account of thestress and strain distributions arising from the DL configuration,obtained by solving Eq. (10). However, in relationship to the

nonlinear behavior of structure, results under LL are obtained bymeans of a restart or a continuation analysis, in which the initialstatus of the structure coincides with the one obtained in the DLconfiguration. The main aim of this step is to evaluate the worststress and displacement effects under LL, i.e. ULS, SLS, and to predictthe new values of hanger, girder and arch cross-sections. Thesolution is performed for a proper number of loading conditions,i.e. NLL, in which, for each bridge components, maximum stressesand displacements are collected, by solving the following discreteequations:

K�ðU�ÞDU

�¼ PLLi � P0ðS�

R; S�L; EG

0 ; EA0 Þ i ¼ 1; . . . ;NLL ð11Þ

where K�is the stiffness matrix of the discrete equations depending

on the current displacement vector U�, P0 is the vector of nodal point

forces arising from the previous increment in element displace-ments and stresses from the dead load to the live load configura-tions, DU

�are the incremental displacement vector and PLLi is the

live load force vector of the current i-th loading combination. Start-ing from results obtained from Eq. (11), the design of arch, girder

D. Bruno et al. / Computers and Structures 170 (2016) 13–25 19

and cable system elements is evaluated by using Eq. (3) or bysolving optimization problem described by Eq. (7). At this stage,the analysis on the cable-system elements can be easily performed

in terms of optimization factors ULðRÞiðjÞ and XLðRÞ

iðjÞ , which are deter-

mined by means of Eqs. (4)–(6) on the basis of maximum stressand displacement values extracted by the LL combinations. Simi-larly, the design of arch and girder cross-sections is here performedby solving the following optimization problems, which are executedseparately on the basis of the maximum stress resultants involvingthe minimum safety factors in the arch or in the girder:

OBJ�1!MinðAGÞGGðAGÞ;CE!D

�G0ðAGÞ6aG;CVs!AGðtG1 ;tG2 ; . . . ;tGnG Þ

OBJ�2!MinðAAÞGAðAAÞ;CE!D

�A0 ðAAÞ6aA;CVs!AAðtA1 ;tA2 ; . . . ;tAnA Þ

ð12Þ

where GG or GA are the global objective functions concerning thevolume involved in the Girder (G) or the Arch (A), D

�G0 or D

�A0 corre-

spond to the design criteria, here introduced as Constraint Equa-tions (CE) for the ultimate limit state design of the girder (G) andthe arch (A) and the sets ðtG1 ; tG2 ; . . . ; tGnG Þ or ðt A1 ; t A2 ; . . . ; t AnA Þ are thecontrol variables of the optimization problem. Moreover, Eqs. (10)and (12) refer to classical optimization problems expressed in termsof minimum displacements or volume minimization involved in thearch or in the girder. The optimum solution is determinediteratively on the basis of previously converged values arising fromthe k � 1 iteration, which are considered as initial values in the nextsubstep, leading to relatively low computational efforts in thesolving procedure.

4. Results

4.1. Analysis for medium span tied-arch bridges

The consistency of the proposed model is investigated for anarch bridge scheme with a medium span length. The main aim ofthe present analysis is to verify, for a bridge scheme involving alow number of variables, the convergence behavior of the iterativeprocedure and the reliability of the optimum configuration. How-ever, subsequently more complex cases concerning long spanbridges will be investigated. The bridge scheme, reported schemat-ically in Fig. 5, presents a total length between vertical supportsand a width equal to 50 m and 10 m, respectively. Moreover, aparabolic profile of the arch, with an aspect ratio H/L equal to0.17 is considered (Fig.5). The arch and the girder are assumed tobe made of S420 steel material (fyk = 420 N/mm2), with a Rectangu-lar Hollow Section (RHS). For the cable prestressing steel, an allow-able stress corresponding to 45% of the ultimate tensile strength,fpk = 1690 MPa, minimum fatigue strength equal to Dr = 200 MPa,modulus of elasticity E = 200 GPa and specific weight c = 77 kN/m3

are considered [20]. The deck has concrete plate and transversebeams, which are assumed to be simply supported by the longitu-dinal edge beams. The hangers are distributed by means of a radialarrangement with a uniform distribution along the arch profile anda constant radial angle / equal to 60� with a spacing step equal to5 m. As a consequence, the total number of elements of the cable

Fig. 5. Synoptic representation and m

system for each arch is 18, i.e. 9 for each orientation. The deadloads concerning the secondary elements or Nonstructural Loads(NSL), due to road pavement, concrete platforms and guardrail,are equal to 80 kN/m and 50 kN/m, respectively. The live loads,defined according to [21], present a transverse distribution basedon three lines of LL, whose equivalent load values, equal to

Qk ¼P3

i¼1Q1ki ¼ 1200 kN and qk ¼P3

i¼1qki ¼ 42 kN=m, are applieddirectly to longitudinal edge beams, considering several positionsto obtain the worst loading scenarios. The design criteria adoptedfor the arch and girder are consistent with prescriptions concern-ing buckling strength analysis arising from the Method II of AnnexD by Eurocodes [19] and those recommended by parametric stud-ies developed in [26]. Finally, only LL concerning traffic loads areconsidered by using factored or unfactored loading combinationsequal to 1.35DL + 1.5NL + 1.5LL or to DL + NL + LL to analyze ULSor SLS/FLS, respectively. However, the generalization of the pro-posed model for considering also the effects of seismic or windforces, can be developed introducing additional loading combina-tions. Moreover, at this stage, not much emphasis was consideredon rigorous recommendations arising from existing design criteria,since the essential aim of the presented investigation is to verifythe efficiency of the optimization algorithm.

At first, convergence behavior is investigated in terms of repre-sentative variables of the cable systems, such as the cross sectionarea of the hangers and the stresses involved in the cables to pre-dict the initial configuration. In particular, in Figs. 6 and 7, for eachiteration, the cross sections of the hangers for the Left (L) or Right(R) orientations are reported as a function of the number of itera-tions (NIT) required to obtain the optimum solution. The analysisdenotes that, with respect to the initial values, the final solutionis strongly modified and presents a convergent evolution towardthe optimum configuration, with a relatively low number of itera-tions. Moreover, the distribution of the cross sections and the ini-tial stresses under DL at the final design configuration are reportedin Fig. 8. The sets of the hangers with left or right orientationsdenote their lowest or largest values in proximity of the archspringing points, respectively, because of the rigidity of the con-nection between arch and tie. Such results are consistent withexisting studies available from the literature on network archbridge design, in which the inefficiency of the lateral cables inthe transferring forces between girder and arch was proved bymany investigations (see for instance [12]). The distribution ofthe cross-section areas is not symmetric due to the presence ofthe simply supported constraint at the right end of the girder.The optimum solution is reached enforcing the lowest materialutilization in the design of the bridge constituents. Such task isanalyzed by the results, described in Fig. 9a and b, in which distri-butions of the optimization factors UR;L, related to design criterionfor representative elements of the cable system from the initial tothe final configurations, are reported. The results denote that allelements of the cable system, in the worst loading scenario, reachthe maximum allowable strength value provided by the designcriterion, i.e. close to the unity. As a consequence, in such configu-ration, the steel quantity, involved in the cable system andpredicted by the optimization procedure, is the lowest possible,

ain data of the structural scheme.

0 2 4 6 8 10 12 140.000

0.025

0.050

0.075

0.250

0.275

0.300

0.325

12

3 4 5 67

89

AL S A

/gL

NIT

(1), (3) (5), (7) (8), (9)

Fig. 6. Convergence behavior of representative normalized Left (L) cross-sectionareas as a function of the number of iterations (NIT).

Fig. 7. Convergence behavior of the normalized Right (R) cross-section areas as afunction of the number of iterations (NIT).

Fig. 8. Distribution of the cross-section areas of the hangers and the stre

20 D. Bruno et al. / Computers and Structures 170 (2016) 13–25

which ensures the best material utilization. The results denote thatthe convergence behavior, in some hanger, appears to be irregulartoward the asymptotic value. Such phenomenon is produced bythe prevalence of the strength criterion utilized in the design ofthe hangers, which can be related to the ULS (maximum stress)or FLS (incremental stress) loading combinations. In order to verifysuch occurrence, in Fig. 9a and b comparisons in terms of maxi-mum values of the performance factors UR;L arising from bothdesign criteria, i.e. ULS or FLS, are reported. The results denote thatfor the internal cables the worst criterion is the one concerning FLSconditions, whereas, for the hangers located at bridge extremities,ULS combinations produce the highest stress values. Moreover, theevolution of the performance factors is not strictly convergent, butit is affected by the prevalence of the worst design criterion, lead-ing to discontinuities in the curves during the iteration steps.

The distribution of the cross-section characteristic lengths, thesteel quantities involved in the main constituents of the bridgeand the corresponding geometric properties are analyzed in Tables2 and 3 and in Fig. 10. In particular, results reported in Table 2show that the evolution of the cross-section lengths and the designcriteria presents an asymptotic convergent behavior toward theoptimum solution. Moreover, the values predicted by the designcriteria in the arch and girder reach the maximum allowable quan-tity, namely close to the unity, ensuring the lowest possible mate-rial quantity involved in such bridge components. From the resultsreported in Fig. 10 or in Table 3, it transpires that the steel quanti-ties, predicted in the arch and girder are quite comparable and aremuch larger than the one involved in the cable system, i.e. withinthe range between 12% and 15%. However, the importance of anaccurate design of the hangers should be considered not only forstructural reasons but also by the larger costs involved in suchelements than those required by girder and arch made of ordinarysteel, typically with ratios in the range between 3 and 7. Resultconcerning geometric properties of the cross-sections denote thatdimensionless axial or bending stiffness ratios, i.e. IG/IA and AG/AA

respectively, differ within a range below than 15% (Fig. 10). Suchresults are quite consistent with structural optimization achieve-ments and current design approach developed in network archbridges, whose general aims are to minimize bending stressesand girder/arch displacements, in such a way to reproduce a trussstructural scheme typically observed in the framework of networksystems. Moreover, since all the members under the worst designscenario reach the lowest margin of safety, the optimum solution

sses under DL as a function of the normalized girder position (X1/L).

Fig. 9. Distribution of the performance factor for the left (a) and right (b) hanger orientation as a function of the number of iterations (NIT) and the design criterion (FLS, ULS).

Table 2Evolution cross section dimensions and the design criteria for the girder (DG) and arch(DA) bridge components as a function of the number of iterations (NIT).

NIT Girder Arch

BG [m] HG [m] tG [m] BA [m] HA [m] tA [m] DG DA

1 1.000 2.000 0.050 1.000 2.000 0.05 0.427 0.9762 0.803 0.931 0.010 0.853 1.232 0.01657 0.878 0.9673 0.797 0.888 0.006 0.835 1.131 0.00702 0.950 0.9534 0.797 0.888 0.006 0.833 1.122 0.00512 0.949 0.9685 0.796 0.884 0.005 0.821 1.060 0.005 0.952 0.9506 0.796 0.883 0.005 0.815 1.033 0.005 0.954 0.9537 0.796 0.883 0.005 0.814 1.024 0.005 0.960 0.9768 0.396 1.185 0.005 0.814 1.024 0.005 1.019 0.9889 0.397 1.188 0.006 0.814 1.024 0.005 0.972 0.998

10 0.407 1.222 0.005 0.814 1.024 0.005 0.978 1.00011 0.409 1.231 0.005 0.814 1.024 0.005 0.981 1.00012 0.421 1.281 0.005 0.814 1.023 0.005 0.981 1.00013 0.421 1.268 0.005 0.814 1.023 0.005 0.981 1.00014 0.421 1.268 0.005 0.814 1.023 0.005 0.981 1.00015 0.421 1.268 0.005 0.814 1.023 0.005 0.981 1.000

Table 3Final values of the design variables related to the hangers.

Left Right

Xi (m) Si (MPa) Ai (cm2) Xi (m) Si (MPa) Ai (cm2)

3.77 268 0.54 6.25 329 8.588.75 147 11.40 11.25 308 4.18

13.75 134 11.30 16.25 237 9.4818.75 155 10.80 21.25 159 9.8323.75 185 10.10 26.25 177 10.1028.75 173 8.72 31.25 148 11.4033.75 235 8.81 36.25 118 13.0038.75 262 5.91 41.25 123 14.1043.75 284 9.83 46.23 250 0.10

Fig. 10. Normalized material steel quantities involved in the Arch (A), Girder (G)and Hangers (H), normalized ratios between cross-section areas (AG/AA) and inertialmoments (IG/IA) as a function of the number of iterations (NIT).

D. Bruno et al. / Computers and Structures 170 (2016) 13–25 21

turns out to be a global minimum in the material quantity involvedin bridge constituents.

4.2. Analysis for long span bridge and comparisons with other bridgeconfigurations

In order to verify the reliability of the proposed methodology,additional results are developed for a more complex bridge case,

involving a total of 180 m span, 25 m width and a rise to span ratioequal to 1/6. In this framework, the analysis is extended also interms of cable system configurations taking into considerationtwo different hanger arrangements, i.e. the network or vertical(see Fig. 11). The main aim of the present results is to verify theconsistency of the optimization algorithm and to propose compar-isons between different bridge typologies. The cable system of thenetwork arrangement is defined by a total number of 70 cables foreach arch and it is based on a constant radial angle / equal to 65�and a spacing step equal to 5 m. Moreover, in order develop a con-sistent comparison between the bridge configurations, the samespacing step is also assumed for the bridge typology with verticalhangers. The main constituents of the girder and arch are madein steel with a RHS (Rectangular Hollow Section), whose geometricvalues are considered by the optimization problem as design vari-ables. Only the arch and girder cross sections enter in the designprocedure and not slab or transverse elements, which connect

Fig. 12. Distribution of the performance factors for the hangers with left orientation(UL) as a function of the number of iterations (NIT). Final distribution of theperformance factors (UL

OPT) as a function of the dimensionless position ofhanger/girder intersection point on the girder (X1/L).

Fig. 13. Distribution of the performance factors for the hangers with rightorientation (UR) as a function of the number of iterations (NIT). Final distributionof the performance factors (UR

OPT) as a function of the dimensionless position ofhanger/girder intersection point on the girder (X1/L).

22 D. Bruno et al. / Computers and Structures 170 (2016) 13–25

the longitudinal beams. However, a generalization of the proposedapproach can be easily developed, also for girder typologies basedon reinforced concrete [15], just introducing the correspondingstrength criterion as required by the optimization algorithm(see Eq. (12)). The dead loads are evaluated during the iterationprocedure because of the geometry changes involved by theoptimization algorithms, whereas the dead loads due to secondarystructural elements or nonstructural loads (road pavement, con-crete platforms and guardrail) are equal to 170 kN/m and 50 kN/m, respectively. Material properties concerning the main structuralelements, i.e. girder, arch and hangers, are assumed to be equal tothe previous investigated case concerning the bridge schemereported in Fig. 5. Similarly, according to the loading combinationdefined in [21], live loads consist of two lines (NL) for each carriage-way and are applied on the width of the bridge, with total loads

Qk ¼P2NL

i¼1Qki ¼ 2000 kN and qk ¼P2NL

i¼1qki ¼ 69 kN=m, to producethe worst effects on the bridge components.

At first, the convergence behavior is analyzed in terms ofoptimization factor evolution for the network system. In particular,in Figs. 12 and 13, the relationships between right and left valuesof UR;L as a function of the number of iterations (NIT) required bythe numerical procedure are presented. Similar results are pro-posed in Fig. 14, in which the distributions of the cross section areain the hangers for a representative number of iterations arereported. The results show that most of the hangers are designedin such a way that, under the worst loading combination, reachexactly the maximum strength, leading to the best performancein the design and the lowest required material involved in thecable system. Moreover, the cross-sections for the left and rightcable orientations denote approximately a symmetric distributionalong the girder projection length, which is basically altered by theconstrain conditions arising at the girder extremities. However, insuch regions, the algorithm predicts the lowest cross-section area,which means that the efficiency of such hangers to the cablesystem transferring forces is practically negligible. Similar observa-tions can be drawn from the results proposed in Fig. 15, in terms ofstress distribution in the hangers under DL and LL configurations.The results show that the prediction of the hanger cross-sectionareas and initial stresses developed by the proposed model are ableto avoid unexpected relaxation effects and thus uncontrollableconcentrations of stress in the adjoining elements. Moreover, stresslevels reached in the hangers, under the worst loading condition,present always a positive value, with margin of safety with respectto unexpected compression status. Results concerning VerticalHanger Arch Bridge (VHAB) schemes are reported in Fig. 16 interms of cable system characteristics, i.e. cross section areas and

Fig. 11. Synoptic representation of the structural scheme.

Fig. 14. Distribution of the cross-section areas of the hangers as a function of thenormalized girder axis (X1/L) and number of iterations.

Fig. 15. Maximum/minimum stresses under ULS and DL conditions.

Fig. 16. VHAB: distribution of the cross-section areas of the hangers and thestresses under DL as a function of the normalized girder axis (X1/L).

Fig. 17. NAB: normalized ratios between cross-section areas (AG/AA) and inertialmoments (IG/IA), worst values of the design criteria in the arch (DA) and girder (DG)as a function of the number of iterations (NIT).

Fig. 18. VHAB: normalized ratios between cross-section areas (AG/AA) and inertialmoments (IG/IA), worst values of the design criteria in the arch (DA) and girder (DG)as a function of the number of iterations (NIT).

Fig. 19. Comparisons between NAB and VHAB: normalized material steel quantitiesinvolved in the Arch (A), Girder (G) and Hangers (H) as a function of the number ofiterations (NIT).

D. Bruno et al. / Computers and Structures 170 (2016) 13–25 23

Table 4Final values of the design variables related to the hangers for the vertical and network arch bridge schemes.

VHAB NAB

Left Right

Xi (m) Si (MPa) Ai (cm2) Xi (m) Si (MPa) Ai (cm2) Xi (m) Si (MPa) Ai (cm2)

5 372 1.66 3.75 303 0.01 6.25 133 9.7710 405 20.40 8.75 377 0.47 11.25 212 14.7015 389 21.20 13.75 394 1.24 16.25 243 14.1020 382 19.60 18.75 393 5.57 21.25 249 12.5025 383 17.60 23.75 387 8.73 26.25 248 11.7030 386 16.30 28.75 382 8.90 31.25 250 12.2035 388 15.50 33.75 374 7.50 36.25 244 12.6040 387 14.90 38.75 367 5.63 41.25 246 12.1045 384 14.20 43.75 361 9.96 46.25 241 11.3050 378 13.30 48.75 355 14.30 51.25 244 10.2055 371 14.30 53.75 353 9.88 56.25 251 11.3060 369 14.80 58.75 353 8.59 61.25 258 12.4065 374 14.10 63.75 345 8.35 66.25 267 12.2070 382 13.50 68.75 337 7.66 71.25 275 10.1075 389 15.70 73.75 336 12.70 76.25 284 9.0680 389 13.80 78.75 333 11.70 81.25 288 10.2085 368 7.15 83.75 329 10.00 86.25 290 10.8090 394 15.40 88.75 326 9.80 91.25 294 9.3495 367 6.85 93.75 322 11.90 96.25 298 7.95

100 389 13.70 98.75 321 11.40 101.25 301 11.20105 390 15.50 103.75 317 10.10 106.25 304 12.50110 387 14.70 108.75 313 10.40 111.25 305 6.84115 387 13.50 113.75 311 13.40 116.25 313 7.24120 387 12.90 118.75 309 13.10 121.25 320 7.66125 388 13.10 123.75 313 12.70 126.25 321 8.93130 389 13.60 128.75 317 11.10 131.25 326 8.32135 389 14.20 133.75 321 13.00 136.25 333 6.60140 388 14.80 138.75 329 13.20 141.25 339 5.21145 387 15.60 143.75 330 14.10 146.25 345 5.14150 384 16.30 148.75 335 13.70 151.25 354 4.25155 382 17.90 153.75 323 14.20 156.25 357 4.10160 382 19.10 158.75 303 14.40 161.25 359 1.84165 388 21.10 163.75 268 17.00 166.25 355 0.70170 400 20.60 168.75 213 17.20 171.25 335 0.06175 367 1.68 173.75 127 7.26 176.25 274 0.01

Table 5Final values of the design variables related to the arch and girder for the vertical andnetwork arch bridge schemes.

Scheme Girder Arch

AG [m2] IGz [m4] IGy [m4] AA [m2] IAz [m4] IAy [m4]

VHAB 0.438 0.373 0.251 0.257 0.0277 0.115NAB 0.088 0.052 0.036 0.2866 0.115 0.0225

24 D. Bruno et al. / Computers and Structures 170 (2016) 13–25

stresses under DL, for different iteration steps. In particular, thedistribution of the hanger cross sections and stresses in the zeroconfiguration is quite smooth and regular, except for the hangerends, because of the girder/arch connections. Such distributionpoints out how in VHAB, the cable system is mainly devoted totransfer the internal forces between arch and girder by means ofuncoupled internal forces. As a matter of fact, in the case of VHAB,the required steel quantity of the cable system is much lower thanone observed for NAB, since in the case of inclined hangers,the cable system contributes notably to the global stiffness of thebridge. Moreover, the shear forces are carried out also by thecable-system and not as in the VHAB by the arch and girder only.In order to verify such concept, results concerning girder and archdimensioning are presented in Figs. 17 and 18, in which the ratiosin terms of cross section characteristics and current values of thedesign criteria for the arch and girder are reported. The analysesdenote that the relative stiffness ratios predicted for the VHABare much larger than those required for the NAB. This result isquite consistent with current studies available from the literature

on arch bridge design, which point out enhanced stiffness proper-ties due to the presence of inclined cables in NAB with respect toconventional bridges based on vertical hanger arrangements [15].Moreover, the proposed algorithm is able to evaluate the optimumconfiguration, in which all the bridge components, i.e. arch, girderand hangers, works under the worst scenarios to the designstrength. Finally, comparisons in terms of steel quantity for theNAB and VHAB structural components are proposed in Fig. 19. Inparticular, the current optimum solution predicted during the iter-ative procedure is described in terms of total material quantityinvolved in girder, arch and hangers. From the results, it emergesthat, in NAB, a lower total steel quantity than the one involved inVHAB is predicted, within a percentage error equal to 43.8%.Although, in the VHAB, arch and cable system present values ofquantity of material lower than the ones predicted in the case ofNAB, i.e. almost 11% and 48% less, respectively, such reductionsof material volume are annihilated by a larger quantity involvedin the girder, which is the most prevalent in the total computationof required volume material. Finally, the final value of design vari-ables concerning the hangers, girder and arch are reported inTables 4 and 5.

5. Conclusions

An optimization model for network arch bridge schemes is pro-posed, in which the optimum solution is achieved by means of aniterative methodology based on three-step algorithm. The pro-posed technique identifies the initial configuration under deadloads in terms of post-tensioning forces in the hangers as well as

D. Bruno et al. / Computers and Structures 170 (2016) 13–25 25

the initial deformations in arch and girder, reproducing the designconfiguration under structural and nonstructural permanent loads.Moreover, the evaluation of post tensioning forces and the opti-mum cross-sectional areas of the bridge components, i.e. archand girder, takes into account of maximum/minimum stress anddisplacement effects produced by the external loads. In the presentoptimization scheme, an elaborate and efficient iteration method-ology is incorporated in a FE code to obtain an approximated solu-tion based on minimum volume criteria. The proposed approachenforces the minimization, which is not exactly imposed as in clas-sical optimization procedure. Actually, the heuristic approach ofthe procedure does not ensure, a priori, that the predicted solutionis a global optima; such configuration can be obtained by means ofcomplex optimization analyses involving all the variables, whichcharacterize the bridge design (geometry, initial stresses, loads,design prescriptions, etc.). However, despite existing methodolo-gies, based on pure optimization procedure, the proposed methodseems to be not affected by numerical convergence problems, sincethe iterative procedure is able to reduce computational efforts, typ-ically documented in standard optimization models. The proposedmethod is able to design each element of the structure in such away that, under the worst loading combination, maximum allow-able strength is achieved, leading to the lowest possible materialutilization. The optimum solution guarantees that, under the worstloading design combinations, the design criterion for each bridgecomponents reaches the maximum allowable value, leading tothe lowest required material quantity involved in the bridge com-ponents. The iterative nature of the proposed methodology onresults arising from dead and live loads, controls the minimumstress level in the hangers, avoiding as a result unexpected relax-ation effects in the cable system elements and amplifications ofthe stress resultants in the adjoining elements or in the arch andgirder. The proposed algorithm is based on a simple procedure,which is based on data easily recoverable by using standard com-mercial FE software packages. Moreover, the proposed procedure isable to solve min/max problem with implicit objective functionals,reaching the global minimum in the feasible set, by means areduced number of iterations. The results show that the cableswhich require the largest values of required steel quantity thanthe rest of the elements are those located at the bridge extremities,almost normal to the geometric axis of the arch; contrarily, thecables approximately parallels denote inefficiency in the transfer-ring forces between arch and girder and thus their contributionis quite negligible. Comparisons in terms of bridge typology denotethat NAB, despite to VHAB, are able to reduce the required materialquantity involved in the bridge components, due to the presenceinclined cables, which improve the global bridge stiffness, leadingto a better distribution of the internal stress resultants in thebridge components.

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