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June 2002

Performance-Based Optimization for Strut-Tie Modeling of Structural Performance-Based Optimization for Strut-Tie Modeling of Structural

Concrete Concrete

Q. Q. Liang University of New South Wales

B. Uy University of New South Wales, brianuy@uow.edu.au

G. P. Steven University of Durham, UK

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https://ro.uow.edu.au/engpapers/187

Recommended Citation Recommended Citation Liang, Q. Q.; Uy, B.; and Steven, G. P.: Performance-Based Optimization for Strut-Tie Modeling of Structural Concrete 2002. https://ro.uow.edu.au/engpapers/187

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Performance-Based Optimization for Strut-Tie Modelingof Structural Concrete

Qing Quan Liang1; Brian Uy, M.ASCE2; and Grant P. Steven3

Abstract: Conventional trial-and-error methods are not efficient in developing appropriate strut-and-tie models in complex structuralconcrete members. This paper describes a performance-based optimization~PBO! technique for automatically producing optimal strut-and-tie models for the design and detailing of structural concrete. The PBO algorithm utilizes the finite element method as a modeling andanalytical tool. Developing strut-and-tie models in structural concrete is treated as an optimal topology design problem of continuumstructures. The optimal strut-and-tie model that idealizes the load transfer mechanism in cracked structural concrete is generated bygradually removing regions that are ineffective in carrying loads from a structural concrete member based on overall stiffness performancecriteria. A performance index is derived for evaluating the performance of strut-and-tie systems in an optimization process. Fundamentalconcepts underlying the development of strut-and-tie models are introduced. Design examples of a low-rise concrete shearwall withopenings and a bridge pier are presented to demonstrate the validity and effectiveness of the PBO technique as a rational and reliabledesign tool for structural concrete.

DOI: 10.1061/~ASCE!0733-9445~2002!128:6~815!

CE Database keywords: Optimization; Concrete structures; Finite element method; Performance.

IntroductionThe shear design of structural concrete members is a complexproblem, which has not been solved fully. The adoption of em-pirical equations in current concrete model codes leads to com-plex design procedures for shear. Empirical equations generallyyield shear strength predictions that deviate considerably fromexperimental results. In addition, they need to be continuouslyevaluated for new materials. These highlight the limitations ofempirical equations and the need for a rational approach to struc-tural concrete. Strut-and-tie modeling has been proved to be arational, unified, and safe approach for the design and detailing ofstructural concrete under combined load effects~ASCE-ACICommittee 445 on Shear and Torsion 1998!. By strut-and-tiemodeling, the influence of shear and moment can be taken intoaccount simultaneously and directly in one model.

Truss models were introduced by Ritter~1899! for the sheardesign of reinforced concrete beams, and extended by Mo¨rsch~1909! to beams under torsion. The truss analogy method receivedconsiderable studies in the 1960s and 1970s~Kupfer 1964; Leon-hardt 1965; Lampert and Thurlimann 1971!. Collins and Mitchell

~1980! proposed the truss model approach that considers defor-mations for design of reinforced and prestressed concrete. It isnoted that truss models can only be used to design regions of aconcrete structure where the Bernoulli hypothesis of plane straindistribution is assumed valid. At regions where the strain distri-bution is significantly nonlinear, the truss model theory is notapplicable. The strut-and-tie model, which is a generalization ofthe truss analogy method for beams, can be used to design dis-turbed regions of structural concrete as demonstrated by Marti~1985!. Schlaich et al.~1987, 1991! extended the truss modeltheory to a consistent strut-and-tie model approach for the designand detailing of any part of reinforced and prestressed concretestructures.

Ramirez and Breen~1991! reported that a modified trussmodel approach with variable angle of inclination diagonals and aconcrete contribution could be used for designing reinforced andprestressed concrete beams. Strut-and-tie modeling has been ap-plied to the design of pretensioned concrete members~Ramirez1994! and posttensioned anchorage zones~Sanders and Breen1997!. Strut-and-tie model approach and related theories for theshear design of structural concrete were summarized in the state-of-the-art report by the ASCE-ACI Committee 445 on Shear andTorsion~1998!. Moreover, the strut-and-tie model design methodhas recently been incorporated in the ACI 318-02 for the designof structural concrete~Cagley 2001!.

Conventional methods for developing strut-and-tie models instructural concrete involve a trial-and-error iterative processbased on the designer’s intuition and past experience. It is a chal-lenging task for the designer to select an appropriate strut-and-tiesystem for a concrete structure with complex geometry and load-ing conditions from many possible equilibrium configurations. Asa result of this, computer programs based on the truss topologyoptimization theory have been developed for generating trussmodels in reinforced concrete structures~Anderheggen and Schla-ich 1990; Ali and White 2001!. Computer graphics as useful de-sign aids have been employed to develop strut-and-tie models in

1Research Fellow, School of Civil and Environmental Engineering,Univ. of New South Wales, Sydney NSW 2052, Australia. E-mail:qqliang@hotmail.com

2Senior Lecturer, School of Civil and Environmental Engineering,Univ. of New South Wales, Sydney NSW 2052, Australia.

3Professor, School of Engineering, Univ. of Durham, Durham DH13LE, U.K.

Note. Associate Editor: Shahram Pezeshk. Discussion open until No-vember 1, 2002. Separate discussions must be submitted for individualpapers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this paperwas submitted for review and possible publication on The manuscript forthis paper was submitted for review and possible publication on July 17,2001; approved on October 9, 2001. This paper is part of theJournal ofStructural Engineering, Vol. 128, No. 6, June 1, 2002. ©ASCE, ISSN0733-9445/2002/6-815–823/$8.001$.50 per page.

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structural concrete~Alshegeir and Ramirez 1992; Mish 1994; Yun2000!. The performance-based optimization~PBO! method forcontinuum structures with displacement constraints proposed byLiang et al.~2000a, 2001! has been shown to be a rational andefficient tool for automatically generating optimal strut-and-tiemodels in reinforced and prestressed concrete structures.

Shape and topology optimization of continuum structure hasbeen reviewed by Hafka and Grandhi~1986! and Rozvany et al.~1995!. Several continuum topology optimization methods havebeen developed in the last two decades. The homogenization-based optimization~HBO! method~Bendso”e and Kikuchi 1988;Suzuki and Kikuchi 1991; Dı´az and Bendso”e 1992; Diaz andKikuchi 1992; Tenek and Hagiwara 1993; Bendso”e et al. 1995;Ma et al. 1995; Krog and Olhoff 1999! treats topology optimiza-tion of continuum structures as a problem of material redistribu-tion within a design domain composed of composite material withmicrostructures. The homogenization theory is used in the HBOmethod to calculate the effective properties of composite material.Simple approaches to topology optimization of continuum struc-tures are also available, such as the density function approach~Mlejnek and Schirrmacher 1993; Yang and Chuang 1994!, thehard kill optimization method~Rodriguez and Seireg 1985; Atrek1989; Rozvany et al. 1992; Xie and Steven 1993!, and the softkill optimization method~Baumgartner et al. 1992!. It should benoted that these continuum topology optimization methods couldlead to many locally optimal solutions. To overcome this prob-lem, performance-based optimality criteria have been proposedby Liang et al.~1999, 2000b,c! and Liang~2001! to identify theglobal optimum as opposed to the local.

This paper extends the PBO method proposed by Liang et al.~2000b! for topology design of continuum structures with meancompliance constraints to the strut-and-tie modeling of structuralconcrete. The development of strut-and-tie models in structuralconcrete is transformed to the topology optimization problem ofcontinuum structures. Optimization criteria for strut-and-tie mod-els are described. An integrated design optimization procedure isproposed for strut-and-tie design of structural concrete. Optimalstrut-and-tie models in a low-rise concrete shearwall with open-ings and a bridge pier are automatically generated by the PBOtechnique, and compared with analytical solutions.

Strut-and-Tie Model Optimization Problem

Strut-and-tie models are used to idealize the load transfer mecha-nism in cracked structural concrete at ultimate limit states. Thedesign task is mainly to identify the load transfer mechanism in astructural member and reinforce the member such that this loadpath will safely transfer applied loads to the supports. In reality,some regions of a structural concrete member are not as effectivein carrying loads as others. By eliminating underutilized portionsfrom a structural concrete member, the actual load path in themember can be found. The PBO method has the capacity to findthe underutilized portions of a member and remove them from themember to improve its performance. Therefore, the strut-and-tiemodeling of structural concrete can be transformed to a topologyoptimization problem of continuum structures.

In nature, loads are transmitted by the principle of minimumstrain energy~Kumar 1978!. Minimizing the strain energy of astructure is equivalent to maximizing its overall stiffness. Thus,strut-and-tie systems in structural concrete should be developedon the basis of system performance criteria~overall stiffness!rather than component performance criteria~strength!. Dimen-

sioning the components of a structural system easily satisfiescomponent performance criteria. It should be noted that the en-hanced ductility design approach should be used to detail thestrut-and-tie model obtained. Based on these design criteria, theperformance objective of the strut-and-tie model optimization isto minimize the weight of a structural concrete member whilemaintaining its overall stiffness within an acceptable performancelevel. For a structural member modeled with plane stress ele-ments, the performance objective can be expressed in mathemati-cal forms as follows:

minimize W5(e51

n

we~ t ! (1)

subject to C<C* (2)

tL<t<tU (3)

whereW5the total weight of a structural concrete member,we5the weight of theeth element;t5the thickness of elements~or thewidth of member cross section!; C5the absolute value of themean compliance of the member;C* 5the prescribed limit ofC;n5the total number of elements in the member;tL5the lowerlimit of element thickness; andtU5the upper limit of elementthickness. To simplify the optimization problem, only uniformsizing of the element thickness is considered in the PBO method.

Limit Analysis and Finite Element Modeling

The behavior of structural concrete members under applied loadscan be well approximated by the uncracked linear, cracked linear,and limit analysis~Marti 1999!. Strength performance predictionsbased on a limit analysis will be reliable if structural concretemembers are designed with adequate ductility and detailing. Thelimit analysis can be divided into lower-bound and upper-boundmethods~Nielsen 1984!. Lower-bound methods require the de-signer to design a structural concrete member by strengthening itsload transfer mechanism. They are particularly suitable for de-signing new concrete structures. On the other hand, upper-boundmethods allow for quick checks for ultimate strength, dimensions,and details of existing structures. They are suitable for the perfor-mance evaluation of existing structures. Strut-and-tie models cor-respond to the lower-bound limit analysis, and can indicate thenecessary amount, the correct locations, and the required detailingof the steel reinforcement.

After extensive cracking of concrete, loads applied to a struc-tural concrete member are mainly carried by concrete struts andsteel reinforcement, which form the load transfer mechanism. Thefailure of a structural concrete member is mainly caused by thebreakdown of the load transfer mechanism, such as the yieldingof steel reinforcement in ductile structural concrete members~ASCE-ACI Committee 445 on Shear and Torsion 1998!. Beforedesigning a structural concrete member, the locations of tensileties and the amounts of steel reinforcement are unknown. Actu-ally, it is the designer’s task to identify an appropriate strut-and-tie system in a structural concrete member in order to reinforce it.As a result of this, the nonlinear behavior of reinforced concretecannot be taken into account in the finite element model for de-veloping strut-and-tie systems.

It is proposed here to develop strut-and-tie systems in struc-tural concrete based on the linear elastic theory of cracked con-crete for system performance criteria and to design the structuralconcrete member based on the theory of plasticity for component

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performance criteria. Only two-dimensional models are consid-ered here. In the finite element analysis, plain concrete membersare treated as homogenization continuum structures, and modeledusing plane stress elements. The PBO algorithm has been writtento link with the finite elementSTRAND6codes~1993! to performthe finite element analysis and optimization tasks in an iterativemanner. The progressive cracking of a concrete member is char-acterized by gradually removing concrete from the member,which is fully cracked at the optimum. It is noted that load-deformation responses of a structural concrete member in an op-timization process are highly nonlinear because the topology ofthe member is changing at each iteration.

Element Removal Criteria

Element removal criteria can be derived by undertaking a designsensitivity analysis on the mean compliance of a structural con-crete member with respect to element removal. A detailed deriva-tion has been given in a previous paper by Liang et al.~2000b!.Element removal criteria are such that elements with the loweststrain energy densities should gradually be removed from the con-tinuum design domain to achieve the performance objective. Thestrain energy density of theeth element is defined as~Liang et al.2000b!

ze5uue

Tkeueu2we

(4)

in which ue5nodal displacement vector of theeth element; andke5stiffness matrix of theeth element.

For a concrete member under multiple load cases, a logicalAND scheme is employed in the calculation of element strainenergy densities for elimination~Liang et al. 2000b!. In the logi-cal AND scheme, an element is deleted from the structural con-crete member only if its strain energy density is the lowest for allload cases. By removing elements with the lowest strain energydensities from a concrete member, the maximum stiffness designat minimum weight can be achieved. In order to obtain a smoothsolution, however, only a small number of elements are removedfrom the discretized concrete member. The element removal ratio

~R! for each iteration is defined as the ratio of the number ofelements to be removed to the total number of elements in theinitial design domain.

Performance-based Optimality Criteria

The performance evaluation of strut-and-tie systems in an optimi-zation process is required in order to determine the optimum. Aperformance index has been proposed by Liang et al.~2000b! forquantifying the performance of bracing systems for multistorysteel building frameworks with an overall stiffness constraint.This performance index is also applicable to strut-and-tie systems,and its mathematical derivation is presented here.

The strain energy or mean compliance of a structure is ex-pressed by

C51

2PTu (5)

Fig. 1. Low-rise concrete shearwall with openings

Fig. 2. Performance characteristics of shearwall with openings

Fig. 3. Performance index history of shearwall with openings

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whereP5nodal load vector; andu5nodal displacement vector.In problems with the element thickness as design variables, an

infeasible design in an optimization process can be converted intoa feasible one by the scaling design procedure~Kirsch 1982;Liang et al. 1999, 2000c!. Since the stiffness matrix of a planestress continuum structure is a linear function of the elementthickness, the element thickness can be uniformly scaled to keepthe mean compliance constraint active at each iteration in the

optimization process. By scaling the initial structural concretemember with a factor ofC0 /C* , the scaled weight of the initialdesign is represented by

W0s5S C0

C* DW0 (6)

where W05the actual weight of the initial design domain; and

Fig. 4. Optimization history of strut-and-tie model in shearwall with openings:~a! topology at iteration 10;~b! topology at iteration 20;~c!toppology at iteration 30;~d! optimal topology at iteration 35;~e! optimal model;~f! model by Marti~1985!

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C05the absolute value of the strain energy of the initial designunder applied loads. Similarly, by scaling the current design withrespect to the mean compliance limit, the scaled weight of thecurrent design at theith iteration can be determined by

Wis5S Ci

C* DWi (7)

in which Ci5the absolute value of the strain energy of the currentdesign under applied loads at theith iteration; andWi5the actualweight of the current design at theith iteration.

The performance index at theith iteration is proposed as

PIES5W0

s

Wis 5

~C0 /C* !W0

~Ci /C* !Wi5

C0W0

CiWi(8)

The performance index is a measure of structural responses andthe weight of a structural member in an optimization process, andthus quantifies the performance of structural topologies. Bygradually eliminating elements with the lowest strain energy den-sities from a concrete member, its performance in terms of theefficiency of material and overall stiffness can be improved. Toobtain the optimal topology, performance-based optimality crite-ria for structures with mean compliance constraints can be pro-posed as

maximize PIES5C0W0

CiWi(9)

The optimal topology obtained represents the most efficient load-carrying mechanism in the continuum design domain. Thus, op-timal topologies generated by the PBO technique can be treatedas optimal strut-and-tie models in structural concrete members.The physical meaning of the performance-based optimality crite-ria is that the optimal strut-and-tie model transmits loads in sucha way that the associated strain energy and material consumptionare a minimum. For a concrete member subject to multiple load-ing cases, the performance index can be calculated by using thestrain energy of the member under the most critical loading casein an optimization process.

Design Optimization Procedure

The design of a structural concrete member using strut-and-tiemodeling involves the estimation of an initial member size, de-

veloping an appropriate strut-and-tie model and dimensioningstruts, ties and nodes. The finite elementSTRAND6codes~1993!are used in the PBO method as a modeling and analytical tool.The PBO algorithm has been written to link withSTRAND6toautomatically carry out the finite element analysis and optimiza-tion tasks. Once the user has set up the finite element model, thecomputer would automatically generate the optimal strut-and-tiemodel. The main steps of the design optimization procedure aregiven as follows:1. Select an appropriate size for the concrete structure based on

serviceability performance criteria and design space con-straints;

2. Model the two-dimensional concrete member using finite el-ement programs. Applied loads, support conditions, and ma-terial properties of the concrete member are specified. Pre-stressing forces can be treated as external loads~Schlaichet al. 1987; Ramirez 1994; Liang et al. 2001!;

Fig. 5. Design domain for bridge pier

Fig. 6. Performance characteristics of bridge pier

Fig. 7. Performance index history of bridge pier

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3. Perform a linear elastic finite element analysis on the con-crete member;

4. Evaluate the performance of the resulting system by usingEq. ~8!;

5. Calculate the strain energy densities of elements for eachloading case;

6. RemoveR ~%! elements with the lowest strain energy den-sities from the concrete member;

Fig. 8. Optimization history of strut-and-tie model in bridge pier:~a!topology at iteration 20;~b! toppology at iteration 40;~c! optimaltopology at iteration 49

Fig. 9. Optimal strut-and-tie model and final design proposal ofbridge pier

Fig. 10. Arrangement of main steel reinforcement in bridge pier

Table 1. Strut and Tie Forces in Strut-and-Tie Model for Bridge Pier

Member number Force~kN!

1 21142 11623 33634 234705 239196 232197 233638 25500

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7. Check model continuity. This is to ensure that the strut-and-tie model generated by the PBO technique is a continuousstructure that satisfies the equilibrium condition;

8. Check model symmetry for a concrete member with an ini-tially symmetrical loading, geometry, and support condition;

9. Save current model. The models generated at each iterationare automatically saved in files for use in a latter stage;

10. Repeat steps 3–9 until the performance index is less thanunity;

11. Select the optimal strut-and-tie model, which correspondsto the maximum performance index;

12. Analyze the discrete strut-and-tie model to determine inter-nal forces in members;

13. Dimension struts, ties, and nodes; and14. Detail steel reinforcement based on the optimal strut-and-

tie model obtained.Dimensioning a strut-and-tie model, which includes sizing the

concrete struts, reinforcing the ties, and checking the bearing ca-pacities of nodal zones, is of significant importance to the overallperformance of a structural concrete member. The detailing ofnodes and steel reinforcement influences the flow of forces in astructural concrete member, and thus directly affects the strengthperformance of concrete struts and ties connected with them. Thekey importance is to ensure that the optimal strut-and-tie modelgenerated by the PBO technique can be realized at ultimate afterdetailing. It should be noted that the optimal strut-and-tie modelproduced by the PBO technique indicates the locations of struts,ties, and nodes but not necessarily their exact dimensions. This isbecause it is developed on the basis of overall stiffness perfor-mance criteria without consideration of strength performance cri-teria. Dimensioning strut-and-tie models should be based on thebearing conditions and strength requirements.

The compressive strength of concrete in struts is influenced byits state of stresses, cracks, and the arrangement of steel reinforce-ment. For safety, the effective compressive strength of concreteshould be used in designing concrete struts. Marti~1985! sug-gested that the effective compressive strength of concrete in strutsshould be taken as 0.6f c8 , whereas Ramirez and Breen~1991!suggested a value of 2.5Af c8~MPa!. Different values of the effec-tive compressive strength of struts could be used in design, de-pending on the state of stresses, cracks, and the arrangement ofsteel reinforcement in the structural concrete member~Schlaichet al. 1987!. Design rules for the effective compressive strengthof struts have been proposed for ACI 318-02~Cagley 2001!.

Reinforcing steel should be provided to carry tensile forces inties in strut-and-tie models. The cross-sectional area of reinforc-ing steel for each tensile tie can be determined from the followingexpression:

f~Asf yr1Apf yp!>T (10)

where f5the capacity reduction factor;As5the cross-sectionalarea of reinforcing bars;f yr5the yield strength of reinforcingbars; Ap5the cross-sectional area of prestressing steel;f p5theeffective yield strength of prestressing steel for tensile ties; andT5the tensile force in a tensile tie.

Illustrative Design Examples

Low-Rise Shearwall with Openings

In this example, the PBO technique is used to automatically gen-erate an optimal strut-and-tie model in a low-rise concrete shear-wall with openings, and numerical results are compared with ana-

lytical solutions. Fig. 1 shows the geometry and loading of alow-rise concrete shearwall with openings based on the examplepresented by Marti~1985!. The shearwall is fixed on the founda-tion. In the present study, the values of the point loadsP1

51,000 kN andP25500 kN are assumed. A compressive cylin-der strength of concretef c8532 MPa, Young’s modulus of con-crete Ec528,600 MPa, Poisson’s ration50.15, and the initialthickness of the shearwallt05200 nm are used in the analysis.The concrete shearwall is modeled using 100-mm-square, four-node, plane stress elements. A mean compliance constraint is con-sidered. The element removal ratioR51% is used.

The performance characteristics of the shearwall in the opti-mization process are presented in Fig. 2. It is seen that by gradu-ally eliminating elements from the shearwall, the mean compli-ance of the shearwall increases with reductions in its weight. Inaddition, rapid increases are observed after more and more ele-ments are deleted from the model. The performance characteristiccurve indicates whether a proposed design for required perfor-mance is feasible. Fig. 3 shows the performance index history ofthe shearwall with openings. It can be seen from Fig. 3 that theperformance of the shearwall in terms of the efficiency of mate-rial and overall stiffness is still gradually improved by eliminatingelements with the lowest strain energy densities from the modeleven if there are a large portion of openings. The maximum per-formance index of 1.2 occurs at iteration 35.

The optimization history of strut-and-tie model in the shear-wall is presented in Fig. 4. When elements with the lowest strainenergy densities are removed from the shearwall, the resultingtopology evolves to a frame-like structure. Fig. 4~d! shows theoptimal topology obtained at iteration 35. This optimal topologyrepresents the load transfer mechanism in the concrete shearwallunder given loading and support conditions, and can be idealizedas the discrete model illustrated in Fig. 4~e!. This model is com-posed only of struts. The optimal strut model of the shearwallwith openings generated by the PBO technique agrees extremelywell with the analytical solution given by Marti~1985!, as shownin Fig. 4~f!.

In detailing the strut-and-tie model, the depths of concretestruts can be based on either the optimal topology shown in Fig.4~d! or the model given in Fig. 4~f!. The final thickness of con-crete struts~or the shearwall! can then be determined by using theeffective compressive strength of concrete based on the forcesthey carry and bearing conditions. Since the strut-and-tie modelobtained has no tensile ties, the main steel reinforcement is notrequired to carry tensile forces in the shearwall. However, a mini-mum amount of steel reinforcement in a form of reinforcingmeshes in compliance with codes of practice should be providedin the concrete shearwall to control cracking, which may be in-duced by shrinkage and temperature effects. For a completed de-sign, the bearing capacities of nodal zones in the model should bechecked.

Design of Bridge Pier

The design domain for a bridge pier is shown in Fig. 5. Thebridge pier fixed on the foundation is required to support fourconcentrated loads of 2,750 kN transferred from four steel gird-ers. An initial thickness 1.5 m is assumed for this bridge pier. ThePBO technique is employed to produce an optimal strut-and-tiemodel for the design and detailing of the bridge pier. A compres-sive cylinder strength of concretef c8532 MPa, Young’s modulusof concreteEc528,600 MPa, and Poisson’s ration50.15 are usedin the finite element analysis. The bridge pier is modeled using

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125-mm-square, four-node, plane stress elements. Plane stressconditions and the mean compliance constraint are considered.The element removal ratioR51% is used in the optimizationprocess.

The performance characteristics of the pier structure in theoptimization process are fully captured by a weight-compliancecurve shown in Fig. 6. Since the overall stiffness of the pierstructure is gradually reduced by element elimination, structuralresponses such as the mean compliance are increased. The topol-ogy performance of the bridge pier in optimization process ismonitored by the performance index shown in Fig. 7. By remov-ing a small number of elements with the least contribution to thestructural stiffness from the pier structure at each iteration, theperformance index increases from unity to a maximum value of1.17. It is observed from Fig. 7 that after iteration 69, the perfor-mance index decreases rapidly. This indicates that further elementelimination leads to the breakdown of the load-carrying mecha-nism in this bridge pier.

Fig. 8 demonstrates the optimization history of strut-and-tiemodel in the bridge pier. It is seen that the load transfer mecha-nism characterized by remaining elements in the pier structurebecomes more and more clear when lowly strained elements aresystematically deleted from the finite element model. The optimaltopology was obtained at iteration 49, as shown in Fig. 8~c!. Ap-plied loads are mainly carried by this optimal structure, whichrepresents the most efficient load transfer mechanism in the de-sign domain considered. The optimal topology shown in Fig. 8~c!is transformed to the discrete strut-and-tie model for the bridgepier illustrated in Fig. 9, where solid lines represent tensile tiesand dotted lines represent compression struts. To achieve betterforce flows within the pier structure and economical designs, afinal design proposal for the bridge pier is presented in Fig. 9. Itis seen from Fig. 8~c! that the pier wall can be designed as twoseparated columns to further improve economical construction.

After the strut-and-tie model has been developed, it is astraightforward matter to dimension it. Forces in members of thestrut-and-tie model shown in Fig. 9 are given in Table 1. It isimportant to provide steel reinforcement to carry tensile forces ininclined tensile ties shown in Fig. 9. The locations of these in-clined tensile ties are difficult to predict by using conventionaltrial-and-error methods~Warner et al. 1998!. An arrangement ofthe main steel reinforcement for resisting tensile forces in thebridge pier is illustrated in Fig. 10. Additional reinforcing meshesthat are not shown in Fig. 10 should be provided in the bridge pierin accordance with the minimum requirements of the codes ofpractice for crack control.

ConclusionsThe performance-based optimization~PBO! technique formulatedon the basis of system performance criteria for automatically gen-erating optimal strut-and-tie models in structural concrete hasbeen described in this paper. Developing strut-and-tie models instructural concrete is transformed to a topology optimizationproblem of continuum structures. Optimal topologies produced bythe PBO technique are treated as optimal strut-and-tie models forthe design and detailing of structural concrete. Performance-basedoptimality criteria for determining optimal strut-and-tie modelshave been developed. An integrated design optimization proce-dure has been proposed for optimizing and dimensioning struc-tural concrete with strut-and-tie systems.

The PBO algorithm has been used to generate optimal strut-and-tie models in a low-rise concrete shearwall with openings and

a bridge pier, and numerical results have been verified by existinganalytical solutions. It has been demonstrated that it is appropriateto develop strut-and-tie systems in structural concrete based onthe linear elastic theory of cracked concrete for overall stiffnessperformance criteria and to design concrete members based on thetheory of plasticity for strength performance criteria. The PBOtechnique presented overcomes the limitations of conventionaltrial-and-error methods for developing strut-and-tie models instructural concrete, and provides concrete designers with an effi-cient automated design tool for complex design situations.

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