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International Journal of Solids and Structures 50 (2013) 121–136

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International Journal of Solids and Structures

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Topology optimization of the fiber-reinforcement retrofitting existing structures

Matteo Bruggi ⇑, Alberto TaliercioDepartment of Structural Engineering, Politecnico di Milano, I20133 Milano, Italy

a r t i c l e i n f o

Article history:Received 23 May 2012Received in revised form 5 September 2012Available online 4 October 2012

Keywords:Topology optimizationFiber-reinforcementOrthotropic materialsTsai–Wu failure criterionStress constraintsMin–max problems

0020-7683/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijsolstr.2012.09.009

⇑ Corresponding author.E-mail addresses: [email protected] (M.

polimi.it (A. Taliercio).

a b s t r a c t

The paper presents a numerical approach for the optimal design of any unidirectional fiber-reinforcementto improve the structural performance of existing structural elements. A problem of topology optimiza-tion is formulated, simultaneously searching for the regions to be strengthened and the optimal point-wise inclination of the reinforcement. Aim of the formulation is the minimization of the maximumequivalent stress in the underlying material, for a prescribed amount of fiber-reinforcement. The Tsai–Wu failure criterion is implemented to detect highly tensile-stressed regions in the existing structuralcomponents, both in case of isotropic material (e.g. concrete) and orthotropic media (e.g. brickwork orreinforced concrete). A suitable set of relaxed stress constraints is dealt with, calling for a no-compressionstress state in the fiber-reinforcement. The resulting multi-constrained min–max problem is solved bymathematical programming. Numerical examples are presented to discuss the features of the achievedoptimal layouts, along with their possible application as preliminary design for structural retrofitting.Performances of the adopted computational procedure are investigated as well.

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1. Introduction

In the last decades the use of fiber reinforced polymers (FRPs)for the retrofitting of existing buildings has become more and morewidespread. Retrofitting can be motivated by the need of meetingcurrent standards, or to protect a damaged structural element fromfurther physical–chemical environmental aggression. The firstapplications of this technique date probably back to the 90s and re-fer to concrete structures (Norris et al., 1997; Triantafillou, 1998).FRP strips or sheets are mostly employed to externally reinforcecracked r.c. beams (Arduini and Nanni, 1997), or to wrap columnsto enhance their mechanical performances under horizontal ac-tions (Shahawya et al., 2000). More recently, externally bondedFRP strips were also employed to retrofit or repair historic masonrybuildings (see e.g. Shrive, 2006). This technique has several advan-tages over standard retrofitting techniques, including flexibility,effectiveness and reversibility. Additionally, in the case of buildingsin seismic regions (see e.g. Garcia et al., 2010), FRP strips do notsignificantly increase the structural mass and the earthquake-induced inertia forces, contrary to conventional techniques suchas external reinforcements with steel plates, surface concretecoatings, and welded meshes. A survey of the applications offiber-reinforced composites in civil engineering can be found inBakis et al. (2002).

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The effectiveness of FRPs in enhancing the mechanical proper-ties of structural elements has been investigated by a number ofresearchers. Among others, El Maaddawy and Sherif (2009) testeddeep beams with openings reinforced by carbon fiber reinforcedpolymers (CFRPs) and reported an increase in bearing capacityranging between 35% and 70% respect to the unreinforced compan-ion beams, according to the location of the openings; an attempttoward the analytical prediction of the shear strength of thereinforced beams was also proposed. Whereas the layout of thereinforcing FRP sheets is quite obvious for r.c. beams in bending(Pisani, 2006), the choice is more complex for beams reinforced inshear. Täljsten (2003), for instance, investigated experimentallythe effect of the orientation of the reinforcing fibers on the strengthof beams wrapped by CFRP fabrics: the highest strength is achievedwhen the fibers are nearly perpendicular to the shear cracks. Simi-lar tests were carried out by Hadi (2003), who focused on the effectsof the material type (CFRP versus E-glass) and the number of layerson the bearing capacity and the ductility of r.c. beams wrapped withFRP sheets. An increasing number of experimental campaigns referto the application of fiber-reinforcement to masonry brickwork, seee.g. Marcari et al. (2007). Because of the importance of this retrofit-ting technique in seismic areas, several authors assessed the perfor-mances of structural elements reinforced with FRP under cyclicloads. This was done e.g. by Garcia et al. (2010), who carried outshaking table tests on full scale r.c. frames with and withoutexternally bonded FRPs at the beam-column joints, by Patersonand Mitchell (2003) and Li and Lim (2010), who cyclically testedr.c. shear walls before and after retrofitting.

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122 M. Bruggi, A. Taliercio / International Journal of Solids and Structures 50 (2013) 121–136

In most applications, the geometry of the reinforcing layers issuggested either by intuition, or by the shape itself of the structuralelement to be retrofitted. Several researchers, however, tried toachieve an optimal design of the FRP reinforcement through a rig-orous approach. Zou et al. (2007), for instance, dealt with the opti-mal design of FRP jackets for the seismic retrofit of r.c. columns byminimizing the volume (hence, the cost) of the reinforcement,using the thicknesses of the jackets as design variables and a recur-sive algorithm. With a similar goal, Perera and Varona (2009) em-ployed a genetic algorithm to optimize the bending and shearreinforcement of r.c. beams.

In the present work, the optimal layout of FRP strips and/orsheets employed to retrofit existing structures is sought adoptinga different perspective, i.e. resorting to methods for the optimaldesign of the topology of the reinforcement. Topology optimizationis a branch of structural optimization that basically aims at estab-lishing the optimal distribution of material within a given designdomain, leading to optimal configurations characterized bymultiply-connected material regions. Since the pioneering contri-bution of Bendsøe and Kikuchi (1988), topology optimization hasbeen mainly concerned with the maximization of the structuralstiffness under volume constraints. The stiffness-based approachhas been extended to achieve different goals, such as maximizingthe fundamental natural frequency or the buckling load, among oth-ers. Refer e.g. to Eschenauer and Olhoff (2001) and Bendsøe andSigmund (2003) for an exhaustive overview on topology optimiza-tion. Optimal design of structures can be found assuming a specificmaterial, that in the case of anisotropy involves orientational design.

In the late 80s and early 90s, the increasing demand for fiberreinforced composite materials has driven extensive research onthe optimal orientation of orthotropic materials in many engineer-ing applications. In the case of bodies made of anisotropic materi-als, such as FRPs, the design variables may include the materialorientation. Three different approaches were originally developedto solve this kind of problems: the strain-based method (Chengand Pedersen, 1997; Pedersen, 1990, 1989), the stress-based meth-od (Cheng and Kikuchi, 1994; Diaz and Bendsøe, 1992; Suzuki andKikuchi, 1991) and the energy-based method (Luo and Gea, 1998).These approaches take into account the effect of the change instrain and stress due to the change in material orientation, to pro-vide effective formulations for the optimal design of structuresmade of orthotropic materials. A general optimality condition thatarises when the structural compliance has to be minimized undergiven static and kinematic boundary conditions, is that the stressand strain tensors must locally share the same principal directions.This result was first obtained by Pedersen (1989) for 2D orthotro-pic solids, and later extended by Rovati and Taliercio (2003) to 3Dorthotropic bodies.

The use of fiber-reinforced polymers in mechanical applicationhas also gained an ever increasing popularity in the optimal designof composite laminate shell structures, with the main goal of deter-mining the stacking sequence by proper choice of material and fi-ber orientation of each FRP layer for any desired structuralperformance, see e.g. Foldager et al. (1998) and Stegmann andLund (2005). In the last decades many efforts have been directedtowards a more general framework that addresses the simulta-neous optimization of material and structure, dealing with theoptimal design of material properties along with its optimaldistribution in the design domain. Within this kind of approachesthe material properties are represented in the most general possi-ble form, i.e. as elements of the unrestricted set of positive semi-definite constitutive tensors of a linearly elastic continuum. Refer,among others, to Bendsøe et al. (1995), Kocvara et al. (2008), andStingl et al. (2009) for further details on this topic.

Most of the formulations for topology optimization deal with theminimization of the structural compliance, interpolating the mate-

rial mechanical properties as smooth functions of the design vari-able, i.e. the material density, through suitable power laws.Reference is made to the so-called SIMP (Solid Isotropic Materialwith Penalization) method, as proposed by Bendsøe (1989). Alter-native approaches have been developed in the literature to take intoaccount the limitation of the effective material strength on the opti-mal topology, based on extensions of the SIMP law to cope withstress constraints, see in particular Duysinx and Bendsøe (1998)and Rozvany et al. (1992). Stress-based design has also been inves-tigated dealing with structural composites, as in the case of the ori-ginal contribution by Swan and Arora (1997). Remaining in the fieldof topology optimization, reference is made to Le et al. (2010) for anextensive review on established methods for strength design,including comparisons among alternative approaches and differentsolving algorithms. Drawbacks of the pure compliance-based opti-mization are generally documented in the case of isotropic materi-als with symmetric behavior in tension and compression, whileonly a few works have recently tackled the case of unequal proper-ties, as in Bruggi and Duysinx (2012) and Luo and Kang (2012),resorting to the adoption of a Drucker–Prager strength criterion in-stead of the conventional von Mises stress measure.

Within the above theoretical framework, this paper investigatesa numerical approach for the topology optimization of the fiber-reinforcement to be applied on existing structural elements inorder to improve their original mechanical performance. Aim ofthe formulation is the minimization of the maximum equivalentstress in the underlying element, for a prescribed amount of fiber-reinforcement. Limited regions to be strengthened are detectedproviding the optimal local orientation of a unidirectional rein-forcement. The Tsai–Wu failure criterion is implemented to searchfor highly tensile-stressed regions in the unreinforced structuralcomponents, thus dealing both with isotropic elements, such asbeams and walls made of plain concrete, and with orthotropicones, such as brickwork or reinforced concrete panels. A suitableset of relaxed stress constraints is dealt with to enforce a purelytensile stress state in the fiber-reinforcement to be arranged overthe structure. The arising multi-constrained min–max problem issolved by mathematical programming, resorting to the Method ofMoving Asymptotes by Svanberg (1987).

The layout of the paper is as follows. In Section 2 the fundamen-tals of the governing equations are reported, introducing the mod-eling of a SIMP-based distribution of unidirectional reinforcementbounded to an underlying structural element. Section 2.2 ad-dresses the Tsai–Wu failure criterion and the related equivalentstress measure for the detection of highly stressed regions in thestructural components to be reinforced. Attention is also paid tothe stress state in the fiber-reinforcement to derive suitableconstraints to deal with the SIMP-based distribution of a no-compression material. Section 3 introduces the proposedmin–max formulation and addresses numerical issues, such asthe strategy adopted to improve the manipulation of the objectivefunction and to relax the constraints on the stress acting in thereinforcement. Section 4 presents numerical examples to discussthe features of the achieved optimal layouts, along with theirpossible application to the preliminary design of any structuralretrofitting. Performances of the adopted computational procedureare investigated as well. Finally, in Section 5 the main findings ofthe work.

2. Governing equations

2.1. Introductory remarks on the reinforcement problem

Within a two-dimensional domain X, consider an underlyinglayer of linear elastic material S, along with a superimposed layer

M. Bruggi, A. Taliercio / International Journal of Solids and Structures 50 (2013) 121–136 123

of linear elastic reinforcement F, see Fig. 1. S denotes the existingstructure to be strengthened, with elastic constitutive tensorCS

ijhkðvÞ ¼ CS0ijhk at any point v 2 X, whereas F denotes an orthotropic

overlying fiber-reinforcement. An orthogonal reference frameOx1x2 is adopted. Let qðvÞ and hðvÞ be two bounded function inX, such that 0 6 q 6 1 and 0 6 h < p, respectively. The former rep-resents the density of the reinforcement in the considered domain;the latter represents the orientation of its fibers with respect to theaxis x1. CF

ijhk ¼ CFijhkðqðvÞ; hðvÞÞ is the fourth order elasticity tensor

for the fiber-reinforcing layer, which depends both on the materialdensity and the orientation at any point v 2 X. Adopting the well-known SIMP model, see e.g. Bendsøe and Kikuchi (1988), one has:

CFijhkðqðvÞ; hðvÞÞ ¼ qðvÞpCF0

ijhkðhðvÞÞ; ð1Þ

where CF0ijhkðhðvÞÞ is the stiffness tensor for the ‘‘virgin’’ medium

depending on hðvÞ and p > 1 is a penalization parameter that is usu-ally assumed to be equal to 3, see e.g. Bendsøe and Sigmund (1999).

Under the assumption of perfect bonding between the underly-ing structure and the overlying reinforcement, both layers sharethe same displacement field u in X, meaning that u ¼ uS ¼ uF .Hereafter, the superscripts (or subscripts) S and F denote quantitiesrelated to the existing structure and to the fiber-reinforcement,respectively. The boundary of the domain, C ¼ Ct [ Cu, consistsof two different parts: the former is subjected to given tractionstS

0, whereas the latter undergoes prescribed displacements uS0.

The weak formulation for the elastic problem under consider-ation can be straightforwardly derived in view of the adoption ofa displacement-based finite element method, and reads: findu 2 H1 such that ujCu

¼ uS0 andZ

XCS0

ijhk þ qpCF0ijhkðhÞ

� �eijðuÞeklðvÞdX ¼

ZCt

tS0 � vdC; ð2Þ

8v 2 H1. In the above equation, compatibility implies that the straineij ¼ 1

2 ðui;j þ uj;iÞ is the same for both structural components, i.e.eij ¼ eS

ij ¼ eFij.

Dealing with a plane stress problem involving two layers oforthotropic materials, the relevant constitutive laws may be spe-cialized in a compact form. To this purpose the strain componentsare re-arranged in an array e ¼ ½e11 e22 2e12�.

If the symmetry axis of the underlying composite are alignedwith the adopted reference frame Ox1x2 throughout X, the arrayof the stress components in the existing structural elementrS ¼ ½rS

11 rS22 rS

12� may be written as:

rS ¼ CS0e; with

CS0 ¼ 11� mS

12mS21

ES11 mS

12ES11 0

mS21ES

22 ES22 0

0 0 GSð1� mS12mS

21Þ

264375; ð3Þ

Fig. 1. Portion of a structural element S retrofitted with a layer of fiber-reinforce-ment F.

where ES11; E

S22 are the Young moduli of the material (along x1 and x2,

respectively), GS is the shear modulus and mS12; mS

21 are the Poisson’sratios; the relationship mS

12ES11 ¼ mS

21ES22 holds.

Analogously, one may define an array of the stress componentsin the overlying composite, that is rF ¼ ½rF

11 rF22 rF

12�. In general, thesymmetry axes of the reinforcing material, say X1 and X2, differfrom the symmetry axes of the underlying material x1 and x2 inX. Hence, a transformation matrix TðhÞ is needed to write the con-stitutive law of the fiber-reinforcement in the reference frameOx1x2, see Fig. 1. With the aim of modeling a reinforcing materialthat exhibits prevailing stiffness along one direction (X1), the elas-tic constants are assumed to vanish but for the Young modulusalong the fiber direction, say EF . Thus:

rF ¼ qp TðhÞCF0TðhÞTe; with

CF0 ¼EF 0 00 0 00 0 0

264375 and T ¼

c2 s2 �2cs

s2 c2 2cs

cs �cs c2 � s2

264375; ð4Þ

where c ¼ cos h and s ¼ sin h. The assumed fiber-reinforcementexperiences, in general, non-vanishing strains along both its sym-metry axes X1 and X2, but the only non-vanishing stress componentis the normal stress along the fiber direction X1, i.e. the first term ofthe array:

erF ¼ qp CF0TðhÞTe; ð5Þ

where erF ¼ ½erF11 erF

22 erF12�; the symbol � denotes quantities referred

to the symmetry axes X1 and X2. The fiber-reinforcement is there-fore modeled as a two-dimensional layer that works as a set ofsprings, since it is able to carry only axial loads in the direction ofthe fibers. A suitable set of constraints will be implemented hereaf-ter, to ensure that the reinforcement acts as a no-compressionmaterial (i.e. erS

11 > 0). This is needed to account for the poorperformance provided by the thin lamina in compression that arewell-known in practical applications. The negligible stiffness undercompressive loads along with the arising of instability phenomenasuggest to exploit fiber-reinforcing strips as distributed ties tostrengthen any brittle structure against potential cracking and ten-sile failure.

Exploiting Eqs. (3) and (4), the weak formulation in Eq. (2) maybe re-written as: find u 2 H1 such that ujCu

¼ uS0 andZ

XCS0 þ qp TðhÞCF0TðhÞT� �

eðuÞeðvÞdX ¼Z

Ct

tS0 � vdC: ð6Þ

8v 2 H1. The bilinear form on the l.h.s. of the above formula allowsthe contribution of each layer of the reinforced structure to theoverall strain energy to be pointed out. In case of no reinforcement,i.e. q ¼ 0, only the existing structural element carries the loads,whereas, for q ¼ 1, the stress state in the underlying element isalleviated depending on the orientation of the reinforcing fibers, h.It is worth remarking that, irrespective of the values of q and h,the energy of the two-layers material remains positive definite.

The discretization of the primal problem in Eq. (6) may bestraightforwardly accomplished through the introduction of theapproximated solution field uh and test field vh, both descendingfrom the adoption of quadrangular elements with bi-lineardisplacement shape functions. A piecewise constant discretizationis adopted for the density field and the orientation field: from hereonwards, xe and te (e ¼ 1; . . . ;N) will denote the values of q and h inthe e-th finite element, respectively. Eq. (6) reduces to thefollowing matrix form:

KS0 þ KFðx; tÞh i

U ¼ F; ð7Þ

where KS0 is the stiffness matrix of the underlying existing structureand KF is the stiffness matrix of the reinforcement depending on the

σLc=5σLt

σLc=5σLt

σLt

σLt

σI

σII

Fig. 2. Parabolic stress failure function in the plane of the principal stresses rI andrII , for uniaxial asymmetry ratio s ¼ rLt=rLc ¼ 1=5.


array of the element densities, x, along with the array of the fiberorientations, t. U stands for the generalized displacement vector,while F is the vector of the nodal equivalent loads. Eq. (7) suggeststhat the considered discrete problem may be also seen as a super-position of two sets of displacement-based plane stress finite ele-ments, denoted by S and F respectively, that share compatibilitythrough their nodal unknowns.

Taking into account the adopted discretization for x and t, Eq.(7) may be re-written as:XN

e¼1

KS0e þ xp

e KF0e ðteÞ

h iU ¼ F: ð8Þ

In the above formula, KS0e is the stiffness matrix of any element of

the existing structure, whereas KF0e is the stiffness matrix of any ele-

ment of the reinforcing layer, that depends on the local orientationof the fibers, te.

2.2. Equivalent stress measures

The Tsai–Wu strength criterion is adopted hereafter to describethe failure of anisotropic materials having different stress limits intension and compression, herein the underlying structure S to beretrofitted. According to Tsai and Wu (1971), the quadraticinequality on the equivalent non-dimensional stress measure req

that defines the feasible domain for an orthotropic compositereads:

req ¼ Fi ri þ Fij rirj 6 1: ð9Þ

The Voigt notation has been used to denote the components of thestress tensor, that is, ri (i ¼ 1; . . . ;6) is the i-th term of the arrayr ¼ ½r11;r22;r33;r23;r13;r12�. Fi and Fij (i; j ¼ 1; . . . ;6) are materialconstants that may be derived from experimental tests. Within theframework of a two-dimensional problem, the equivalent stressmeasure defined in above equation may be specialized in terms ofthe only nonzero stress components as:

req ¼ F1 r1 þ F2 r2 þ F11 r21 þ F22 r2

2 þ 2F12 r1r2 þ F66 r212: ð10Þ

The constants Fi; Fij in the above equation may be expressed interms of compressive strength values rL1c and rL2c along the direc-tions 1 and 2, respectively, of the relevant tensile strength values,rL1t and rL2t , and of the shear strength rLs:

F11 ¼ �1

rL1t � rL1c; F22 ¼ �

1rL2t � rL2c

;

F1 ¼rL1t þ rL1c

rL1t � rL1c; F2 ¼

rL2t þ rL2c

rL2t � rL2c;

2F12 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiF11F22

p; F66 ¼

1r2

Ls

:

ð11Þ

In case of an isotropic medium with unequal behavior in tensionand compression, one has rLc ¼ rL1c ¼ rL2c and rLt ¼ rL1t ¼ rL2t:the Tsai–Wu criterion in Eq. (10) may be reduced to a parabolicstrength criterion setting F66 ¼ 3=ðrL1t � rL1cÞ, see e.g. Taliercio andSagramoso (1995). Under this assumption Eq. (10) may bere-written in terms of the first stress invariant J1 and the secondinvariant of the deviatoric stress J02 and the resulting quadraticconstraint on req reads:

req ¼3

rLc � rLtJ02 þ

rLc � rLt

rLc � rLtJ1 6 1: ð12Þ

Fig. 2 shows the above strength represented in the plane of theprincipal stresses rI and rII for rLc ¼ 5rLt . Feasible stress statescorrespond to points falling within the domain bounded by thedepicted curve, or on the boundary of the domain itself, meaningthat the inequality of Eq. (12) holds. For rLc ¼ rLt the equivalentvon Mises stress, depending only on J02, is recovered.

The Tsai–Wu failure function provides a smooth scalar measurethat may be conveniently adopted to detect highly-stressed re-gions in the existing structure to be reinforced, both in the caseof isotropic elements, as for components made of plain concrete,and in the case of orthotropic media, as for brickwork or panelsmade of reinforced concrete. These materials exhibit a brittlebehavior in tension, since their asymmetry ratio s ¼ rLt=rLc is gen-erally less than 1/15. Fiber-reinforcement is therefore applied as anadditional tensile layer that has the aim of relieving the regionswhere req attains the highest values within the domain.

Solving Eq. (8) one may fully recover the kinematics of the prob-lem and compute the relevant component of the strain tensor efrom the displacement field at equilibrium. Recalling Eq. (3), thestress tensor in the layer S modeling the unreinforced structuremay be written in Voigt form as rS ¼ CS0e in the reference frameOx1x2. At a specific point of the e-th finite element, e.g. the cen-troid, one may compute the array of the stress components asrS

e ¼MS;eU, being MS;e the ‘‘stress matrix’’ of the element made ofthe underlying material and U the generalized displacement vectorthat solves Eq. (8), see e.g. Duysinx and Sigmund (1998). Accordingto Eq. (10), the equivalent Tsai–Wu stress measure for the the layermodeling the existing structure in the e-th finite element reads:

rSe;eq ¼ rS

e

� �tF11 F12 0F12 F22 00 0 F66

264375rS

e þF1

F2

0

264375rS

e: ð13Þ

Resorting to some algebra, that is here omitted for the sake of brev-ity, one may write:

rSe;eq ¼ Ut

e Q S;e Ue þ LS;e Ue; ð14Þ

where Q S;e and LS;e are ‘‘failure stress matrices’’ that recover thequadratic and the linear part of the Tsai–Wu equivalent stress forthe existing structure, in the e-th finite element, from the general-ized displacement vector.

Similarly, Eq. (5) allows the components of the stress tensor inthe fiber-reinforcement F to be expressed as erF ¼ qp CF0TðhÞTe,referring to its symmetry axes X1 and X2. At a relevant point ofthe e-th finite element, e.g. the centroid, the stress array in thereinforcing layer can be computed as erF

e ¼ xpe MF;eðteÞU, being

MF;eðteÞ the ‘‘stress matrix’’ of the element made of the overlyingmaterial that depends on the fiber orientation te. The only non-vanishing term in erF

e is the normal stress along the fiber directionX1 (see Section 2.1). The equivalent stress measure for the layermodeling the fiber-reinforcement in the e-th finite element maybe therefore simply computed as:

rFe;eq ¼ ½1 0 0�t erF

e ; ð15Þ

meaning that:


rFe;eq ¼ xp

e LF;eðteÞUe ¼ xpe r

Fe;eqðteÞ; ð16Þ

where LF;e is a ‘‘failure stress matrix’’ depending on te, that recoversan equivalent stress for the fiber-reinforcing layer, in the e-th finiteelement, from the generalized displacement vector.

The above equation gives the macroscopic stress erFe in any ele-

ment of the fiber-reinforcement, of density xe. Following Duysinxand Bendsøe (1998), an appropriate failure criteria for the porousSIMP material should be defined on the so-called apparent ‘‘local’’stress herF

ei that may be derived as herFei ¼ erF

e=xqe , with q > 1. A suit-

able form to enforce a no-compression stress state in the e-th finiteelement of the fiber-reinforcement, in conjunction with the SIMPmodel, reads:

hrFe;eqi ¼

rFe;eq

xqe¼ xðp�qÞ

e rFe;eqðteÞP 0; ð17Þ

where hrFe;eqi is the equivalent ‘‘local’’ stress measure in any finite

element of the reinforcing layer.

3. The topology optimization problem

3.1. Problem formulation

Established approaches to the reinforcement of certain regionsof concrete structural members adopt the strut-and-tie model(STM) to recover the load-carrying system following the minimumstrain energy principle. STMs for the ultimate limit state designmay be generated through topology optimization searching for stifftruss-like structures whose ties call for the disposal of tensile rein-forcement, see e.g. Liang et al. (2001, 2000) and Bruggi (2009). Thestatic theorem implies that the energy-based STM provides a lowerbound of the load-carrying capacity of an elastic–plastic structuralmember if the statically admissible truss does not violate the yieldcriterion, as also experimentally investigated by Schlaich et al.(1987). The required ductility assumption is generally fulfilled incommon concrete structures, e.g. providing ad hoc distributedreinforcement, but becomes questionable if energy-based STMsare applied to brittle materials such as masonry.

Conventional formulations for the minimization of the elasticstrain energy generate equilibrated truss-like designs and, gener-ally, do not include an explicit enforcement of full consistence ofthe achieved STM with the stress regime in the original structure.Among the others, Moen and Guest (2010) show that auto-equilibrated compression-only loads may originate strut-onlydesigns, without providing the expected reinforcement that shouldbe placed along the principal tensile directions of the structuralelement to cope with the arising force spreading. Recalling thatin traditional r.c. design steel bars are positioned where tensilestresses are found, Biondini et al. (2001) implemented an ad hocstress-constrained minimum weight formulation for truss designto generate stress-path adapting STMs for both the serviceabilityand the ultimate limit state.

Differences between energy-based and stress-based topologyoptimization have been thoroughly investigated in the literature,see e.g. Duysinx and Bendsøe (1998) and Rozvany (1998) forcomparisons of volume-constrained compliance minimization ver-sus stress-constrained volume minimization and Li et al. (1999) fora discussion on the equivalence between stress criterion and stiff-ness criterion in evolutionary structural optimization. For the sameamount of material, similar results are found if the members of theoptimal layout exhibit a homogeneous state of stress and apressure-independent strength criterion is adopted in the mini-mum volume problem. Differences arise if a non-symmetricstrength criterion is implemented in stress-based simulations,since conventional energy-based approaches are not sensitive to

an unequal failure behavior in tension and compression, see e.g.Bruggi and Duysinx (2012).

While the strut-and-tie method finds equilibrated truss-likepaths that fully replace a two-dimensional domain in carryingthe loads, the approach proposed herein provides optimal layersof reinforcement applied over the existing structure to help it incarrying the external forces. A stress-based approach is able toprovide a retrofitting model that is consistent with the stressregime of the underlying existing structure and takes into fullaccount the strength criteria governing its load-carrying capacity.An optimal reinforcement is tailored to strengthen either a plainmaterial, concrete or masonry, or a structural element with anembedded reinforcement, as found in case of an existing reinforcedconcrete member to be retrofitted.

Aim of the proposed approach is distributing a limitedamount of fiber-reinforcement, that has to be properly orientedin order to minimize the maximum value of the equivalentTsai–Wu stress measure rS

eq over the existing structure. Accord-ing to the theoretical framework introduced in Section 2, the dis-crete stress-constrained formulation can be cast in the followingform:

minxe ;te

maxe¼1;N

rSe;eq

n os:t:

XN

e¼1

KS0e þ xp

eKF0e ðteÞ

h iU ¼ F;

XN

xeVe

XN

Ve

,6 Vf ;

xðp�qÞe rF

e;eqðteÞP 0; for e ¼ 1; . . . ;N

0 6 xe 6 1;0 6 te < p:

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

ð18Þ

As already mentioned, the objective function of the above prob-lem is the maximum equivalent stress computed in all the N ele-ments into which the existing structure is subdivided, accordingto Eq. (14). This gives rise to a so called min–max problem,whose unknowns are the array of the element densities, xe, andthe array of the element orientations, te, for the fiber-reinforcinglayer; their ranges of variation are defined in Eqs. (18.5) and(18.6). Eq. (18.2) enforces the discrete equilibrium equation dis-cussed in Section 2.1, whereas Eq. (18.3) represents a constrainton the available amount of reinforcement. The structural weightof the fiber layer is computed multiplying the element densityxe by the relevant volume Ve over the N elements in the mesh,and is required to be less than a prescribed percentage of thewhole design domain, say Vf . The set of constraints in Eq.(18.4) enforces a no-compression stress state in the fiber rein-forcement, by constraining the equivalent ‘‘local’’ stress measurehrF

e;eqi defined in Eq. (17).Effects of the tuning of the parameter Vf on the optimal layouts

will be further investigated in the numerical section. As expected,different assumptions on the allowed volume fraction lead todifferent kind of solutions, meaning that the formulation in Eq.(18) is not able to provide a global optimum that is independenton Vf . This limitation may be straightforwardly overcome resortingto the ‘‘performance index’’ introduced by Liang and Steven (2002),who tackle the problem of improving the performance of acontinuum design domain both in terms of the efficiency of mate-rial usage and overall stiffness. Alternatively, one could reformu-late Eq. (18) in order to minimize the amount of reinforcementsubject to constraints on the maximum equivalent stress that isallowed in the existing layer. The optimal selection of topologyfor the minimum-weight design with stress constraints may beaccomplished according to Liang et al. (1999).


3.2. Numerical implementation

The problem in Eq. (18) is handled via mathematical program-ming, resorting to the Method of Moving Asymptotes (MMA)(Svanberg, 1987) as minimizer. As detailed in the algorithm notes,the min–max problem may be straightforwardly solved writing aminimization problem where the constraints in Eq. (18.2–4) arepreserved, along with the optimization unknowns and their rangeof variations, while a new scalar objective function is considered,say z 2 R. To recover a formulation that is fully equivalent to theoriginal one, an additional set of constraints must be imple-mented, calling for z P rS

e;eq, for e ¼ 1; . . . ;N. Reference is alsomade to Brittain et al. (2012) for a discussion on alternativeapproaches for the solution of min–max problems arising intopology optimization.

According to the adopted solution strategy, the problem in Eq.(18) is turned into a standard multi-constrained minimization thatis tackled with sequential convex programming, see e.g. Duysinxand Bendsøe (1998). To update both xe and te, the gradient-basedminimizer MMA is provided, at each step, with the current valuesof the sets of constraints on rS

e;eq and rFe;eq along with their sensi-

tivity information. The optimization procedure is stopped if therelative difference between the values of every density or orienta-tion unknowns at two subsequent iterations is less than a giventolerance, herein taken equal to 10�3. To improve performanceof the computation, restricted sets of constraints are passed tothe minimizer, see e.g. Haftka and Gürdal (1992), adopting strate-gies that depend on the stress measure to be controlled. An effec-tive control on the highly-stressed regions of the underlyingexisting structure may be performed passing to the minimizer avery limited set of rS

e;eq-values, instead of the whole set of N en-tries. The largest NS

a ¼ 50� 100 values evaluated at each algo-rithm’s step have been used in the simulations of Section 4 formeshes of about 2000� 8000 elements. The achieved results arethe same as for wider sets of evaluation, but the implementedselection has allowed a remarkable amount of computational timeto be saved while preserving good convergence features. In theconsidered retrofitting problems, the elements that are responsi-ble for the tensile failure of the underlying structure are locatedin limited regions of the domain to be reinforced and their set re-mains substantially unchanged during the optimization process.Referring to the constraints on rF

e;eq, an alternative strategy hasbeen implemented. In the performed numerical simulations, onlythe NF

a constraints in Eq. (18.3) whose l.h.s. is larger then 0.65are considered to be active during the first iteration. The thresholdis linearly increased until the tenth step and is constantly set to0.85 thereafter, see also Bruggi and Duysinx (2012). The investiga-tions performed in Section 4 will show that a very limited set ofactive constraints is able to steer the minimizer towards optimalsolutions that are free from the arising of undesired compres-sive-stressed regions of reinforcement. It must be remarked thatthe proposed selection strategies acting on rS

e;eq and rFe;eq introduce

discreteness into the problem and cause a loss of consistency withthe related sensitivity calculation. This may give raise to oscilla-tory behavior in more complicated structures. In such cases theadoption of selection parameters accounting for larger sets of con-straints is suggested.

When enforcing constraints on the fiber-reinforcement layer,one has to account for the singularity problem that may cause con-vergence to undesired local minima with extended gray regions,thus preventing the minimizer from finding the expected pure0–1 design, see e.g. Cheng and Guo (1997). This has been shownto depend on the asymptotic behavior of the ‘‘local’’ stresses inEq. (17), that remain finite and nonzero for a vanishing materialdensity xe ! 0 when p ¼ q is assumed to preserve physical consis-tency (Duysinx and Bendsøe, 1998). A classical way out to the

problem consists in the adoption of ad hoc mathematicalrelaxations to improve the convergence of the minimizer. Thenumerical simulations presented hereafter resort to the alternativeqp-approach (Bruggi, 2008; Bruggi and Duysinx, 2012), instead ofthe classical e-relaxation (Cheng and Guo, 1997). The adoption ofan exponent q < p in the selected Eq. (18.3) provides a strongrelaxation of the constraints in the low density region while intro-ducing no remarkable bias at full density. The assumption q ¼ 2:8for a SIMP exponent p ¼ 3:0 will be adopted in Section 4, unlessdifferently specified.

It is worth recalling that the adoption of a bi-linear interpolationfor the displacement field, along with an element-wise discretiza-tion of the density, may lead to numerical instabilities, such asthe checkerboard problem, as outlined e.g. in Sigmund andPetersson (1998) with reference to conventional problems oftopology optimization. The reinforcement formulation in Eq. (18)is expected to undergo similar instabilities, as it basically aims atdistributing a certain amount of material over a given underlyinglayer. Filtering schemes may be adopted as a classical remedy,which also allows for a heuristic control over the mesh-dependencyof the minimum thickness of the members that arise in the optimallayout. For alternative methods refer e.g. to Sigmund (2007), Guest(2009), and Guest et al. (2004). This work implements an approachbased on the density filter, see Bourdin (2001) and Bruns andTortorelli (2001), transforming the original density variables xe intoa new set of physical unknowns ~xe. Since the e-th finite element hastwo unique design variables, i.e. the density xe and the fiber orien-tation te, the filtering technique is applied to define also a new set ofphysical unknowns ~te. One has:

~xe ¼1PNHej

XN

Hejxj; ~te ¼1PNHej

XN

Hejtj;

Hej ¼X

N

maxð0; rmin � distðe; jÞÞ;ð19Þ

where distðe; jÞ is the distance between the centroid of the e-th andj-th element, respectively, and rmin > dm is the filter radius, being dm

the reference length of the finite element edges. Unless differentlyspecified, in the following numerical simulations rmin ¼ 1;5dm.

3.3. Sensitivity computation

At each iteration of the minimization procedure, the sensitivitycomputation of two sets of equivalent stress measures must beperformed, namely the values rS

e;eq, computed in the NSa elements

experiencing the highest stresses in the underlying existing struc-ture, and the values rF

e;eq, computed in the NFa elements selected for

the imposition of the no-compression constraints in the overlyingfiber-reinforcing layer. The number of stresses to be managed, i.e.NF

a þ NSa, is generally smaller than the number of design variables,

i.e. N, thanks to the adopted selection strategy. The adjoint methodis therefore advantageously implemented to perform the sensitiv-ity computation both for xe and te, see Duysinx and Bendsøe (1998)and Duysinx and Sigmund (1998).

Let consider first the equivalent stresses rSe;eq in the e-th finite

element of the underlying existing structure, Eq. (14). Its deriva-tives with respect to the unknowns xk and tk can be computedavoiding the explicit evaluation of the terms @U=@xk and @U=@tk,through the solution of an adjoint equality that recovers theequilibrium Eq. (7). After some algebra one has:

@rSe;eq

@xk¼ �eUT @KF

@xkU and

@rSe;eq

@tk¼ �eUT @KF

@tkU;

where :

KS0 þ KFðx; tÞh ieU ¼ Q S;eUe þ Q t

S;eUe þ LS;e;

ð20Þ

Table 1Examples 1–5. Elastic and strength constants adopted in the simulations to model the underlying existing structure according to Eqs. (3) and (11), respectively.

Material model (label) ES11 (MPa) ES

22 (MPa) GS (MPa) mS12

rL1t (MPa) rL1c (MPa) rL2t (MPa) rL2c (MPa) rLs (MPa)

‘‘Plain concrete’’ 21,000 21,000 8677 0.21 1.50 20 � rL1t rL1t rL1c ðrL1t � rL1c=3Þ1=2

‘‘Masonry’’ 8530 8030 2580 0.19 0.15 20 � rL1t rL1t rL1c ðrL1t � rL1c=20Þ1=2

‘‘Reinforced concrete’’ 21,000 21,000 8677 0.21 1.50 20 � rL1t 20 � rL1t rL1c ðrL1t � rL1c=3Þ1=2

Fig. 3. Examples 1–3. Geometry and boundary conditions. Dimensions in m.

Fig. 4. Example 1. Contour plot of the equivalent stresses rSe;eq for the ‘‘plain

concrete’’ element to be fiber-reinforced.


meaning that each one of the NSa measures selected in the active set

of stresses to be minimized requires the solution of one additionalload case for the linear system in Eq. (18.2).

The sensitivities of the equivalent stresses rFe;eq, required by the

constraints of Eq. (18.4), call for some additional computational ef-fort. Looking at Eq. (17), the derivatives of the ‘‘local’’ stress mea-sure for the e-th finite element hrF

e;eqi with respect to xk and tk

may be straightforwardly written as:

@hrFe;eqi

@xk¼ dekðp� qÞxp�q�1

e rFe;eq þ

@rFe;eq

@xkxp�q

e ;

@hrFe;eqi

@tk¼@rF

e;eq

@tkxp�q

e ;

ð21Þ

being dek the Kronecker delta. Hence, one may write the sensitivitiesof the entry rF

e;eq with respect to the unknowns xk and tk exploitingEq. (16). As seen before, the computation of the terms @U=@xk and@U=@tk may be overcome through the solution of an adjoint equa-tion recovering the elastic equilibrium, i.e.:

@hrFe;eqi

@xk¼ �eUT @KF

@xkU;

@hrFe;eqi

@tk¼ �eUT @KF

@tkUþ @LF;eðteÞ

@tkUe;

where :

KS0 þ KFðx; tÞh ieU ¼ LF;eðteÞ:

ð22Þ

Again, each one of the NFa active stress constraints on the fiber-

reinforcing layer requires the solution of one additional load casefor the linear system in Eq. (18.2). Eventually, it is remarked thatthe adoption of the filtering approach introduced in Section 3.2implies a chain rule modification of the sensitivities of the full setof stress measures.

4. Numerical simulations

The proposed formulation for the optimal reinforcement istested adopting simplified models for the underlying structuralelements that are commonly encountered in the applications.Table 1 reports elastic and strength properties, as required by thestress–strain relation in Eq. (3), and by the Tsai–Wu stress measurein Eq. (11). Referring to the layer of fiber-reinforcement, athickness thF ¼ 0:15 mm is adopted unless differently specified,

whereas the elastic modulus EF is taken equal to 230,000 MPa,see Section 2.1.

The optimization is performed starting from an initial guessthat consists of a homogenous layer of reinforcement arrangedover the structure, i.e. ~xe ¼ Vf everywhere. Fibers are initially ori-ented according to the local direction of the largest principal stressin the unreinforced existing material. The starting condition on ~te isinspired by practical application, since fiber-reinforcing strips aregenerally distributed as ties carrying axial loads in the directionof the principal tensile stresses in the structural material to be ret-rofitted. The initial guess on ~xe provides the minimizer with a fea-sible solution, since it fulfils the volume constraint in Eq. (18.3).

4.1. Example 1. Square lamina subject to shear loads

Fig. 3(a) shows a square lamina, subjected to self-equilibratedshear loads, say f ¼ 10 kN/m, along the whole external perimeter.An isotropic ‘‘plain concrete’’ is assumed as material model for thestructure to be reinforced, see Table 1. The strength parameters areselected so as to define a parabolic stress criterion with an asym-metry ratio s ¼ rLt=rLc ¼ 1=20. The presence of an inner squarehole with 90� corners generates highly-stressed regions in thevicinity of the geometry singularities, which may be highlightedplotting contours of the equivalent stress measure rS

eq of Eq. (13).The regions experiencing high tensile stresses, arising aroundtwo opposite corners, are highlighted in red in Fig. 4. The blue

Fig. 5. Example 1. Optimal fiber-reinforcement for Vf ¼ 0:15: topology (a) and orientation (b).

Fig. 6. Example 1. Difference between the optimal orientation of the reinforcingfibers for Vf ¼ 0:15 and the direction of the maximum principal stress in theunreinforced structural element.

10 20 30 40 50 60 700.86

0.88

0.9

0.92

0.94

0.96

0.98

1Convergence

Iteration

Obj

ectiv

e fu

nctio

n

thF = 0,150 mmthF = 0,300 mm

Fig. 7. Example 1. Optimal fiber-reinforcement for Vf ¼ 0:15: convergence plots fordifferent values of the fiber-reinforcement thickness.


regions are mainly compressed, and are not deemed to be danger-ous for a material with the considered uniaxial asymmetry ratio.

The stress state highlighted above is often responsible forcracks that arise in the vicinity of tensile-stressed corners inconcrete panels with holes. To avoid this phenomenon, onemay implement the formulation in Eq. (18) looking for the opti-mal distribution and orientation of a fiber-reinforcement layerwith allowed volume fraction Vf ¼ 0:15. Fig. 5 shows the optimaldesign, along with the relevant orientation of the tensile-stressedfibers. Throughout this section, red marks stand for tensile-stressed fibers, whereas blue marks refer to compressive regions.The limited amount of material is located around the holecorners, where rS

e;eq attains its highest values, whereas the orien-tation of the fibers is nearly homogeneous within the bulk ofboth reinforcement regions. In practical applications, fiber-reinforcement is usually applied in the form of straight strips,which are basically subjected to uniaxial tension. Fiber-reinforcing strips may be superposed, to provide either stiffnessand strength along different orientations or increased thickness,as required. The optimal reinforcement found in Fig. 5 suggestsa practical arrangement of the fiber-reinforcement that mainlyconsists of two strips, of suitable width, arranged at 45� to theedges of the design domain.

It must be remarked that the optimal fiber-reinforcement orien-tation is related to the direction of the tensile principal stress in theunderlying structure, since fibers help the brittle unreinforcedstructure to withstand tensile stresses. The optimal orientation,however, is not necessarily the same as that of the largest principalstress in any structural element, since it mainly originates from theelastic equilibrium of a two-layered anisotropic structure, see Sec-tion 2.1. Fig. 6 shows contours of the change in the orientation ofthe fibers at the optimum with respect to the unreinforced ‘‘plainconcrete’’ element. Variations of about 20� are detected in the bulkregions of the optimal reinforcement.

Fig. 7 shows the smooth convergence plot for the objectivefunction of Eq. (18.1), i.e. the evolving maximum value of theequivalent stress measure rS

eq for the selected set of NSa ¼ 50 ele-

ments on a mesh of about N ¼ 2050. The o.f. is normalized to themaximum value evaluated in the unreinforced structure. The de-crease in terms of maximum equivalent stress between the unrein-forced and the retrofitted structure depends of course on thethickness of the fiber-reinforcing layer. Fig. 7 shows that doublingthe reinforcement, i.e. arranging two overlapped strips of thicknessthF ¼ 0:15 mm each, the decrease in the maximum value of rS

eq isnearly twice that provided by the single-layered solution. Therelevant design has been omitted since it is nearly the same aspresented in Fig. 5.

A further investigation is performed on the same example for alarger volume fraction, i.e. Vf ¼ 0:45. Fig. 8(a) and (b) shows theoptimal design that is achieved by the proposed formulation ifthe no-compression constraints in Eq. (18.4) are disregarded. Mostof the distributed reinforced material transfers tensile stresses be-tween adjacent loaded edges of the domain, but a fraction of theoptimal fiber-reinforcement is found in the vicinity of the

Fig. 8. Example 1. Optimal fiber-reinforcement for Vf ¼ 0:45: topology and fiber orientation without constraints on the stress state of the fibers (a,b) and for a conventionalno-compression reinforcement (c,d).


compressed corners of the central hole. This reinforcement locallyincreases the stiffness of the existing structural element, thusrelieving the overall stress level. Fig. 8(c) and (d) shows thetension-only design that is found by the original completeformulation in Eq. (18). As seen in the case Vf ¼ 0:15, the achievedlayout is well suited for a straightforward disposal of FRP strips,due to the nearly homogeneous orientation of the fibers in mostregions of the domain. Fig. 9 plots contours of the difference be-tween the optimal orientation of the fibers and the direction ofthe tensile principal stress in the unreinforced element. Except

Fig. 9. Example 1. Difference between the optimal orientation of the reinforcingfibers for Vf ¼ 0:45 and the direction of the maximum principal stress in theunreinforced structural element.

for limited regions next to the geometrical singularities, thehighest differences are reported in the vicinity of the external sidesof the design domain.

In the case Vf ¼ 0:45, the optimization procedure arrangesfiber-reinforcement so as to connect contiguous loaded edges ofthe domain, thus providing an effective transfer path for the tensilestresses. This result recalls solutions achieved by topologyoptimization methods to automatically derive load paths andconsequently spot out concrete members which can be interpretedas STMs, see Section 3.1. A tie-only retrofitting layout arises in theproposed reinforced problem, because compressive stresses arecarried by the existing structure.

Fig. 10(a) compares the convergence plots achieved in the caseVf ¼ 0:45, with or without enforcing the no-compressionconstraints on rF

eq. As expected, the complete formulation in Eq.(18) provides an optimal value of the objective function that isslightly higher than the unconstrained solution. Fig. 10(b) showsthat a very limited number of local enforcements NF

a is found tobe active during the optimization of the fiber-reinforcement layer.This set is able to steer the solution towards the distribution ofno-compression material in the very first steps of the procedure,without slowing down convergence with respect to the uncon-strained formulation. In Fig. 10(b) the influence of the penaltyfactor q for the apparent local stress on the number of constraintsactive during the optimization procedure is also shown. If q ¼ 2:5the algorithm calls for about 20 active constraints in the first 10iterations and provides the same solution that is achieved for themore demanding default value q ¼ 2:8. Both diagrams highlightthat the adopted selection strategy remarkably reduces the

20 40 60 80 100 120 1400.9

0.92

0.94

0.96

0.98

1

Iteration

Obj

ectiv

e fu

nctio

n

Convergence

without constraints on fibers’ stresswith constraints on fibers’ stress

10 20 30 400

5

10

15

20

25

30

35

Num

ber o

f act

ive

cons

train

ts

Constraints on fibers stress

Iteration

q = 2,8q = 2,5

Fig. 10. Example 1. Optimal fiber-reinforcement for Vf ¼ 0:45: convergence with and without constraints on the stress state of the fibers (a) and number of active localenforcements for different values of the parameter q for the stress-constrained formulation (b).


computational burden with respect to a multi-constrained settingworking on the full set of enforcements for rF

e;eq. The same tension-only design is achieved processing Eq. (18) for NF

a ¼ NSa ¼ N,

meaning that accuracy of the solution is preserved. The consideredformulation distributes retrofitting material over an underlyingfixed layer whose stress regime governs the problem. Thisincreases the robustness of the adopted selection strategies withrespect to standard stress-constrained topology optimization cop-ing with full material and void.

Two additional simulations are performed for Vf ¼ 0:45, adopt-ing a finer mesh of about N ¼ 8000 elements. Fig. 11(a) shows theresult achieved implementing a filtering procedure with a radiusrmin ¼ 3:0dm, whereas Fig. 11(b) reports the optimal layout foundby an unfiltered optimization (see Section 3.1). As expected, theproposed framework is affected by the arising of checkerboard pat-terns for the density unknowns xe. No equivalent instability isfound on the fiber orientations te since they are strongly relatedto the principal tensile direction of the existing structure, seeFig. 9. The adoption of the filtering scheme detailed by Eq. (19) isan effective remedy to recover the optimal solution already pre-sented in Fig. 8. The same filter radius is implemented on both setsof unknowns to preserve consistence with the theoretical frame-work introduced in Bourdin (2001) that implements filteredvariables as new physical unknowns governing the pointwiseconstitutive law of the material. Additionally, this has the aim of

Fig. 11. Example 1. Optimal fiber-reinforcement for Vf ¼ 0:45 on a larger m

smoothing variations in the distribution of the fiber orientation,thus improving readability of the optimal layouts in the arrange-ment of fiber-reinforcing strips.

4.2. Example 2. Beam subjected to distributed vertical loads

The example deals with the ‘‘plain concrete’’ beam representedin Fig. 3(b), subject to a vertical load f ¼ 100 kN/m that is evenlydistributed along the upper side. Only half of the domain is mod-eled in the simulations, exploiting the symmetry of geometryand boundary conditions. Three configurations, differing in termsof position ‘‘a’’ of the two pointwise supports, are investigated toappreciate the effect of the beam slenderness on the optimal rein-forcement for Vf ¼ 0:15.

Fig. 12 shows results referring to the case a ¼ 1:75 m. The opti-mal layout achieved without enforcing the set of no-compressionconstraints on rF

e;eq is presented in Fig. 12(a) and (b). Regions inthe lowest part of the domain are relieved by the presence of atensile reinforcement, whereas the increased stiffness providedby the straight compressive strip at the upper edge improves theoverall stress regime. To obtain an optimal reinforcement consist-ing of fibers subjected only to tensile stresses, the full formulationin Eq. (18) is implemented, and the simplified layout of Fig. 12(c) isfound. As expected, the removal of the compressive reinforcementis matched by a wider reinforced area at the lower midspan. The

esh: topology achieved with filter radii rmin ¼ 3:0dm (a) and rmin ¼ 0 (b).

Fig. 12. Example 2. Optimal fiber-reinforcement for a ¼ 1:75 m and Vf ¼ 0:15: topology and fiber orientation without constraints on the stress regime of the fibers (a,b) andfor a conventional no-compression reinforcement (c,d).


optimal fiber orientation straightforwardly inspires an effectiveretrofitting that mainly consists of three FRP strips: one parallelto the lower edge, and two inclined approximately of 45� withrespect to the beam axis (Fig. 12(d)). It is worth noticing that thehorizontal strip provides a reinforcement against the maximumprincipal stresses, as suggested by the bending stress diagram.The V-shaped reinforcement transfers a fraction of the appliedvertical load to the upper part of the domain and somehow hangsit to the arch-like compressed zone running through this region.

Two alternative configurations are investigated. The pictures inFig. 13(a) and (b) refer to the assumption a ¼ 0:75 m. For thisslender beam the layout shown in Fig. 12(c) is not optimal anymore. The map of the fiber orientations still shows that someinclined reinforcement is required, but the optimal arrangementof the reinforcing layer mainly consists of a longer horizontal stripthat has an increased thickness with respect to the previous case.The pictures in Fig. 13(c) and (d) refer to the case a ¼ 2:25 m, forwhich the highest tensile bending stresses are expected at theupper edge of the domain, in correspondence of the supports. Ateach critical region, the optimal solution couples a short horizontal

Fig. 13. Example 2. Optimal fiber-reinforcement for Vf ¼ 0:15: topolo

reinforcement with an upturned V-shaped arrangement of thestrips. In this case the inclined reinforcements act like ties thathang the outermost portions of the beams as they were cantileversclamped at the support sections. The optimal inclination of the V-shaped reinforcement obviously depends on the distance betweenthe supports.

4.3. Example 3. Panels subjected to shear loads

The square domain depicted in Fig. 3(c), subject to shear loadsf ¼ 100 kN/m at the lower and upper edges, is reinforced for anallowed volume fraction of fiber-reinforcement Vf ¼ 0:45. Thevertical displacements of the loaded edges are constrained, mean-ing that the panel carries stresses due to a combination of shearand bending. A first investigation is performed accounting for theisotropic ‘‘plain concrete’’ model considered in the previousexamples. The achieved optimal design is shown in Fig. 14(a),and is characterized by two extended reinforced regions, locatedat opposite sides of the panel, where fibers share a nearly homoge-neous orientation of 45� with respect to the horizontal direction.

gy and fiber orientation for a ¼ 0:75 m (a,b) and a ¼ 2:25 m (c,d).

Fig. 14. Example 3. Topology and fiber orientation of the optimal fiber-reinforcement for Vf ¼ 0:45: ‘‘plain concrete’’ panel (a), ‘‘masonry’’ panel (b) and ‘‘reinforced concrete’’panel (c).


The achieved result suggests to place the optimal reinforcementmainly in the form of adjacent strips along the diagonal of the pa-nel. Larger regions must be reinforced in the vicinity of the groundconstraints.

An additional simulation is performed on the same geometryconsidering the orthotropic ‘‘masonry’’ model in the second rowof Table 1. Elastic properties are derived from data available inthe literature, see e.g. Anthoine (1995). Strength parameters forthe Tsai–Wu criterion have been taken equal to those used previ-ously, but for the shear strength of the underlying structure. Dueto the assumed reduction in terms of rLt , loads acting upon thestructure have been accordingly scaled of a factor 10 when tacklingthe ‘‘masonry’’ model. Fig. 14(b) shows the achieved optimal de-sign, that is very similar to that of Fig. 14(a). In the central regionof the panel the optimal reinforcement for ‘‘masonry’’ is slightlythicker than the one obtained for the ‘‘plain concrete’’ model. Thisis in agreement with the assumed decrease in terms of shearstrength rLs.

Eventually, the ‘‘reinforced concrete’’ material model n the sec-ond row of Table 1 is implemented. Stiffness and strength param-eters are those of the ‘‘plain concrete’’ model adopted in the firstsimulation, with the only exception of the tensile strength alongthe vertical direction, rLt2, that is taken equal to 20 � rLt1. Thisassumption has the aim of providing a rough modeling of concreteelements where vertical steel bars supply a significant increase intensile strength in that direction compared with ‘‘plain concrete’’.In this example the normal bending stresses in the existingstructure are carried by the vertical steel bars, whereas the fiber-reinforcement has mainly to bear the tangential stresses due toshear loads. This case means to be representative of practicalsituations where concrete panels provided by a classical detailingof vertical steel bars must be retrofitted to behave as shear walls

in seismic areas. Fig. 14(c) shows the achieved optimal design, thatis remarkably different from those obtained in the previous cases.An optimal layout for the fiber-reinforcing strips consists of ele-ments placed along the diagonal of the shear wall, matching theproposals found in many papers on seismic retrofitting, see e.g. Liand Lim (2010). Additionally, in the vicinity of the corners, someminor regions where the reinforcing fibers are parallel to the hor-izontal and vertical directions are found at the boundaries of themain tie. This helps transferring tensile actions from the unrein-forced regions of the underlying structure to the main diagonal tie.

Fig. 15 investigates the effect of the adoption of the three mate-rial models on the optimal orientation of the fibers with respect tothe direction of the tensile principal stress in the unreinforcedstructural elements. As one may easily see, the ‘‘plain concrete’’and ‘‘masonry’’ model have many similarities. Conversely, if theparameters governing the strength model are strongly anisotropic,the orientation of the principal maximum stress in the unrein-forced structure sensibly differs from the optimal orientation ofthe fibers. Fig. 15(c) reports maximum variations of about 50�, con-firming that an effective optimization strategy can not be limitedto simply placing fiber-reinforcement along the principle tensilestresses of the structure to be retrofitted.

4.4. Example 4. Panel with an opening subjected to vertical loads

This example deals with the panel made of ‘‘plain concrete’’ rep-resented in Fig. 16(a). The panel is clamped at the lower edge,whereas a vertical evenly distributed load f ¼ 50 kN/m acts uponthe upper side. A wide central opening weakens the wall and in-duces a remarkable perturbation with respect to the nearly homo-geneous vertical flux of compressive stresses that would cross asolid panel. Tensile stresses arise at the lower and upper sides of

Fig. 15. Example 3. Variations in the optimal layout of the fibers for Vf ¼ 0:45 with respect to the tensile principal directions of the unreinforced structural element: ‘‘plainconcrete’’ panel (a), ‘‘masonry’’ panel (b) and ‘‘reinforced concrete’’ panel (c).

Fig. 16. Examples 4 and 5. Geometry and boundary conditions.


the opening, due to a beam-type behavior that affects both regions,see Fig. 17(a). The problem has also been investigated in the exper-imental literature, testing the effectiveness of different fiber-reinforcement layouts depending on the dimension of square orrectangular openings made in walls of different geometries, seeEl Maaddawy and Sherif (2009) and Mohammed et al. (2010). Con-ventional solutions resort to the adoption of FRP strips placedalong the edges of the openings. These are often coupled with rein-forcements inclined at 45� to resist the effect of horizontal actions(see also Fig. 5).

Fig. 17(b) shows the optimal layout achieved through theformulation in Eq. (18), for Vf ¼ 0:15. The orientation of the fibersis approximately the same as that of the FRP strips that can beidentified in the presented picture. The upper architrave is relieved

Fig. 17. Example 4. Contour plot of the equivalent stress rSe;eq for the ‘‘plain concrete’’ e

by a V-shaped reinforcement scheme, similar to that shown for thebeam in Fig. 12(b). This intercepts most of the vertical load andhangs it up to the outer continuous portions of the wall, actingas columns. Additionally, a short horizontal strip can help thearchitrave in bearing the bending stress arising at midspan dueto the remaining part of the load. An arch-like stress path arisesin the lower part of the panel. The abutments are in the vicinityof the lower corners of the opening, while the key is located atthe midpoint of the constrained edge of the wall. In fact, the rein-forcing strip acts as a tie to equilibrate the arising horizontalthrusts.

4.5. Example 5. Shear-resisting architrave

The last example deals with the optimal reinforcement of anarchitrave connecting two vertical elements and subjected to hor-izontal actions, see Fig. 16(b). A distributed shear load f ¼ 50 kN/macts upon the structure.

A first investigation is performed adopting the ‘‘plain concrete’’material model for the equivalent stress measure rS

eq plotted inFig. 18(a). According to Fig. 18(b), the optimal arrangement ofthe FRP strips can be straightforwardly identified. Similarly tothe previous example, the optimal fiber direction is approximatelyparallel to the FRP strips that can be identified by the map of theoptimal distribution of reinforcement. Inclined strips strengthenthe critical corners of the architrave and span over the verticalelements up to the outer edges. There, the orientation of thereinforcement is mainly vertical and extended up to the ground

lement to be fiber-reinforced (a) and optimal fiber-reinforcement for Vf ¼ 0:15 (b).

Fig. 18. Example 5. Contour plot of the equivalent stress rSe;eq for the structural element to be reinforced and optimal fiber-reinforcement for Vf ¼ 0:15: ‘‘plain concrete’’

element (a,b) and ‘‘reinforced concrete’’ element (c,d).

10 20 30 40 50 60 700.88

0.9

0.92

0.94

0.96

0.98

1Convergence

Iteration

Obj

ectiv

e fu

nctio

n

"plain concrete""reinforced concrete"

Fig. 19. Example 5. Optimal fiber-reinforcement for Vf ¼ 0:15: convergence plotsfor ‘‘plain concrete’’ and ‘‘reinforced concrete’’ elements.


constraints, thus resisting the normal bending stresses. Minor re-gions of reinforcement are found along the inner edges of the ver-tical elements subjected to tensile stresses.

A final investigation is performed on the same geometry assum-ing the material to obey the ‘‘reinforced concrete’’ model. Account-ing for the strength anisotropy of the underlying structure, thenormal bending stresses arising in the vertical elements are notcritical any more for the equivalent stress measure rS

eq shown in

Fig. 18(c). The optimal result represented in Fig. 18(d) suggeststhe adoption of an inclined reinforcement strip running throughthe architrave and connecting the critical corners shared with thevertical elements. Remark that the regions to be strengthened arenot limited to highly stressed zones, but extend into the bulk of thecoupled elements to provide an effective path for the tensile stres-ses arising from the applied shear loads. This is in agreement withresults of the theoretical and experimental literature of aseismicdesign, see e.g. Park and Paulay (1975).

Fig. 19 shows the convergence plots concerning the aboveoptimal designs. In both cases the optimizer solves the multi-constrained min–max problem achieving optimal results througha smooth convergence. As one may easily see, the anisotropicTsai–Wu model implies an increased computational cost withrespect to the parabolic strength criterion.

5. Conclusions

An original approach based on topology optimization has beenpresented to cope with the optimal design of any unidirectionalfiber-reinforcement retrofitting existing structural elements. For aprescribed amount of strengthening material, the formulation spotsout regions to be retrofitted and defines pointwise the optimalinclination of the fiber-reinforcement. A stress-constrained min–max problem has been formulated searching for the optimalreinforcing layer that minimizes the maximum equivalent stressof the underlying existing structure while calling for a no-compression stress state in the fibers. The Tsai–Wu failure criterion


has been implemented to detect highly tensile-stressed regionsboth in case of isotropic material (e.g. concrete) and orthotropicmedia (e.g. brickwork or reinforced concrete). Suitable selectionstrategies have been proposed and tested to improve the numericaltractability of the arising multi-constrained problem that is tackledthrough sequential convex programming. Numerical simulationsshow that a limited set of equivalent stress measures govern theoptimization, while a few constraints are needed in the very firstiterations of the procedure to effectively steer the solution towardsa no-compression reinforcement.

Solutions found by the implemented algorithm are shown tomatch some literature results about structural retrofitting. Theachieved results may inspire enhanced arrangements of the FRPstrips, whose optimal orientation is not necessarily the same asthat of the maximum principal stress in the underlying existingstructure. For a low volume fraction, the optimal reinforcementhas mainly to be placed according to a suitable layout thatstrengthens the regions where the highest equivalent stresses arefound. Additionally, it may help the underlying structure in reliev-ing the stress transferring part of the applied loads out of thecritical zones (see Examples 2, 4 and 5). For increased amount ofmaterials, the formulation arranges fiber-reinforcement so as toconnect contiguous loaded edges of the domain, thus providingan effective transfer path for the tensile stresses (see Examples 1and 3). Differently from a conventional strut-and-tie model, theproposed procedure provides a tension-only retrofitting layoutthat aims at releasing stresses carried by the existing structure.The optimal reinforcement is tailored depending on the strengthcriteria that governs the structural element and is found to playan important role in the definition of the retrofitting solution.

It must be remarked that the proposed algorithm assumesperfect bonding between the underlying structural layer and theoverlying retrofitting. An anchorage length must be providedfollowing the prescriptions given by the technical codes, assimilarly detailed for steel bars in reinforced concrete. In case ofhighly-stressed composite strips, one should also account for thepossibility of delamination. According to international standards,this can be managed prescribing an additional control on the stressin the fiber-reinforcement. Incidentally, note that this prescriptioncould be straightforwardly embedded in the proposed formulationby implementing an additional set of constraints on rF

eq, ascurrently under investigation.

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