Homogenization models for nonlinear and limit analysis of FRP-strengthened masonry

Chapter 16

Homogenization models for nonlinear and limit analysis of FRP-strengthened masonry G. Milani1, A. Chiozzi2 and A. Tralli2 1

Department of Architecture, Built Environment and Construction Engineering (A.B.C.), Technical University of Milan, Milan, Italy, 2Department of Engineering, University of Ferrara, Ferrara, Italy

16.1 Introduction As witnessed by recent seismic events, the inadequate performance of masonry constructions due to the low resistance of masonry under horizontal loads is a well-known matter for all technicians and practitioners involved in the safety assessment of historical city centers (Moon et al., 2007; Ramos and Lourenc¸o, 2004; Yi et al., 2006). Traditional retrofitting techniques, such as the use of external steel plates or reinforced concrete coatings, have shown to be uneconomical, inefficient, and often nonviable. Moreover, they present significant shortcomings mainly due to the additional mass put on the structure. The need for designing efficient and noninvasive strengthening interventions thus appears to be one of the open challenges to be resolved by engineers involved in the repair and/or rehabilitation of masonry buildings before and after earthquakes. In the last 30 years, FRP strengthening emerged as an interesting solution for masonry upgrading, because the technique is able to substantially improve the load-bearing capacity of brickwork structures. It is also a very light solution that provides both high mechanical strength and resistance to corrosion, and is available in a wide set of different commercial forms like laminates, textiles, and tendons (Korany et al., 2001; Corradi et al., 2002; Korany and Drysdale, 2007). Most common structural units, such as planar walls or arches and vaults, can be suitably retrofitted through the application of composite materials on their surface (both intrados and extrados). The most important effect of a generic strengthening intervention with FRP is indeed to preclude the formation of the failure mechanism that Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00016-6 Copyright © 2019 Elsevier Ltd. All rights reserved.

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causes the collapse of the nonstrengthened structure (Foraboschi, 2004). The objective is the formation of a new collapse mechanism different from the unstrengthened case, with higher internal dissipation. In parallel to the practical application of FRP strengthening, robust and advanced nonlinear numerical tools able to give predictions beyond the linear elastic range on the behavior of FRP-strengthened masonry with any shape and under various loading conditions have been developed during the last two decades (Luciano and Sacco 1998; Caporale et al., 2006; Milani and Lourenc¸o, 2012, 2013a; Basilio et al., 2014; Bertolesi et al., 2016a). Up to 15 years ago, research was mainly focused on the mechanics of FRP retrofitted-reinforced concrete elements (Saadatmanesh, 1994), but especially in the last decade, a considerable body of work, both theoretical and experimental, has addressed FRP application to masonry as well (Luciano and Sacco, 1998; Marfia and Sacco, 2001; Valluzzi et al., 2001; Aiello and Sciolti, 2006; Grande et al., 2008; Fagone and Briccoli Bati, 2008; Cancelliere et al., 2010; Fedele and Milani, 2011; Oliveira et al., 2011; Ghiassi et al., 2012). Furthermore, the scientific community has recently put considerable effort into devising simple mathematical relations that allow practitioners to correctly tackle the design of FRP-strengthened masonry structures (see, e.g., Triantafillou and Fardis, 1997; Triantafillou, 1998a). Finally, Italian Guidelines (CNR-DT200, 2013) proposed a simplified but sound interpretation of the phenomenon of delamination that is applicable at the design level. Ideally, to be fully predictive, a numerical model should take into account a number of important structural aspects, exhibited by strengthened masonry at low levels of the external loads and at the verge of collapse. These aspects include: G

G

G

The low masonry resistance against tensile stresses, due to the insufficient capacity of mortar joints to behave elastically in the tension range. The orthotropy in both the elastic and inelastic range (Lourenc¸o, 2000; Massart et al., 2004; Calderini and Lagomarsino, 2008; Plevris and Asteris, 2014). Orthotropy is significantly related to the texture of the masonry, both for in- and out-of-plane actions. For horizontal stretching and horizontal bending, that is, out-of-plane flexion with rotation along a vertical axis (Cecchi et al., 2005; Milani et al., 2006a; Milani and Lourenc¸o, 2010; Mercatoris and Massart, 2011; Milani and Taliercio, 2015, 2016; Bertolesi et al., 2016b), the masonry texture produces perceivable effects that tend to become more evident with the progressive degradation of the material. The different topology of the continuous horizontal mortar joints with respect to the vertical joints, interrupted by the blocks, implies that the shear response of the mortar plays a key role. FRP delamination (see, e.g., Luciano and Sacco, 1998; Triantafillou, 1998b; Marfia and Sacco, 2001). Delamination is typically brittle and

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depends on many concurring factors, such as material and adhesive bulk properties, surface conditions, possible chemical
physical treatments before the FRP application, and environmental conditions (temperature and humidity during and after the strengthening intervention). Conversely, to be efficient, a structural model should avoid a micromodeling representation, which would require prohibitive computational costs. As a consequence, a suitable way for the analysis of FRP-strengthened walls could be a technique based on homogenization applied in two steps (see, e.g., Milani et al., 2009; Milani and Tralli, 2011; Bertolesi et al., 2016b; Milani and Bertolesi, 2017a). In such a procedure, the first step relies on rigorous or simplified homogenization of nonstrengthened masonry, assuming as unit cell either a curved or flat representative volume element (depending on if the strengthened structure is vaulted or not), which furnishes either stress
strain homogenized nonlinear relationships or homogenized strength domains in the case, respectively, of nonlinear or limit analyses. The second step is the structural analysis (limit analysis if the interest of the user is to predict collapse loads and failure mechanisms or incremental nonlinear within a displacement-based design) performed with suitable numerical tools on the already homogenized material where FRP is applied externally. The present chapter reviews two recently presented homogenized approaches belonging to the aforementioned family of two-step procedures, namely a limit analysis and an incremental nonlinear finite element (FE) procedure. In particular, the first approach discussed is an adaptive homogenized upper bound limit analysis, particularly indicated for the prediction of the load-carrying capacity of curved structural elements reinforced with FRP. As a matter of fact, masonry vaults constitute one of the prevailing structural types in historical environments worldwide (Como, 2013; Chiozzi et al., 2016b). Therefore, interest in their safeguard through FRP noninvasive strengthening has increased exponentially. When fast estimates of the increase in load-carrying capacity and immediate insight into the efficiency of a strengthening intervention (intended as the ability to change the failure mechanism) are needed, limit analysis is the most suitable tool to consider. Typically masonry structural elements are studied at failure as assemblages of rigid bodies where dissipation is allowed only on interfaces and— to some extent—such an approach fits well with FE upper bound limit analyses of cohesive-frictional materials, where dissipation is also allowed between contiguous elements (Sloan and Kleeman, 1995; Tralli et al., 2014). Nevertheless, when FE limit analysis models made by rigid and infinitely resistant elements where plastic dissipation is allowed only on interfaces, are adopted, while the number of variables is hugely reduced, the collapse mechanism is constrained to run exclusively along interfaces. It is certainly true that such an approach is in agreement with the actual behavior at failure of

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masonry (especially if made of bricks characterized by good strength properties) that exhibits collapse mechanisms characterized by large blocks mutually roto-translating, but the numerical problem turns out to be highly meshdependent, with the probable outcome of considerable error in the evaluation of the load-bearing capacity. For this reason, the alignment of interfaces becomes pivotal and a FE procedure will perform very poorly if an inappropriate mesh is adopted. An adaptation of the mesh is therefore paramount and, in this framework, one of the authors of the present chapter proposed recently to iteratively adjust mesh nodal points through a sequential programming linear scheme (Milani and Lourenc¸o, 2009; Milani, 2015a). Here, a more advanced procedure, more suitable for FRP-reinforced vaults, is reviewed (see, e.g., Chiozzi et al., 2016a, 2017a,b,c). Vault geometry is described by a nonuniform rational b-spline (NURBS) representation of the midsurface, which can be generated within any commercial-free form modeler, together with information about the local thickness at each point of the surface. By exploiting the properties of NURBS functions, a mesh of the given surface made of very few elements, which still provides an exact representation of the vaulted surface, can be obtained. Therefore, a given masonry vault with any geometry can be represented by few NURBS parametric elements, each element of the mesh being a NURBS surface idealized as a rigid infinitely resistant body. FRP reinforcement, typically made of long composite strips in adherence with the underlying masonry material, is modeled through NURBS as well. Such elements are characterized by both FRP
masonry interfaces and FRP
FRP interfaces. An upper bound limit analysis problem with very few optimization variables is thus deduced, approximating closely the actual geometry of the vault, but using very few variables. Due to the very limited number of rigid elements used, the quality of the collapse load thus found depends on the shape and position of the interfaces, which are adjusted using a genetic algorithm (GA) (Milani and Milani, 2008). The second part of this chapter is devoted to a review of a recently presented homogenized nonlinear FE model suitable for the nonlinear static analysis of FRP-strengthened masonry structures of any shape. From a convenient homogenization approach performed at the mesoscale, macroscopic nonlinear stress
strain relationships are obtained and implemented in a structural nonlinear FE code for a realistic step-by-step analysis of FRPstrengthened masonry flat and curved structures beyond the linear elastic range. Rigid infinitely resistant wedge-shaped 3D elements interconnected by nonlinear interfaces are utilized to model homogenized masonry at the structural level. The utilization of 3D elements is suitable to simulate the flexural strength (Korany and Drysdale, 2007; Mosallam, 2007) increase obtained by the introduction of FRP strips. On the other hand, wedge-shaped elements are preferred with the aim of reproducing possible diagonal out-of-

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plane failures, due to the development of cracks (caused by bending and torsion) that zigzag between contiguous bricks. FRP strips are modeled using triangular rigid elements. Masonry and FRP layers interact using interfacial tangential actions between triangles (FRP) and wedges (masonry). Furthermore, possible limited tensile strength for the FRP strengthening is considered at the interfaces between adjoining triangular elements. Since delamination is a typical fragile phenomenon, elastic behavior followed by a degradation of strength until zero in correspondence of a predefined slip is assumed in the structural scale problem, following formulas provided by the recent Italian norm CNR-DT 200 (Milani et al., 2006b; Fedele and Milani, 2010, 2012; CNR-DT200, 2013; Basilio et al., 2014; Bertolesi et al., 2018). In this way, delamination phenomenon at the FRP
masonry interface and FRP tensile failure may be taken into account as well. The incremental nonlinear elasto-plastic discretized problem is solved at each step using a simple sequential quadratic programming procedure, after having suitably approximated the nonlinear stress
strain relationships of the materials using linear piecewise constant functions. Both the adaptive limit analysis and the incremental nonlinear FE approach are benchmarked on a variety of examples of practical interest. When dealing with limit analysis, a straight and skew masonry arch, both reinforced with longitudinal FRP strips and a masonry dome with annular strengthening, are analyzed. The skew arch is particularly intriguing for the activation of a torsional hinge as a consequence of the asymmetry in the geometry, whereas the straight arch example is discussed to investigate the performance of the NURBS model in a case where more straightforward procedures are typically effective. As far as the second approach is concerned, a deep beam strengthened with FRP strips is analyzed for masonry loaded in-plane. Additionally, a circular arch with buttresses is considered to benchmark the capabilities of the model for curved structures. For the examples presented, both the nonstrengthened and FRP-strengthened case are discussed and detailed comparisons with experimental data are provided. In order to further assess reliability, results obtained through alternative nonlinear FE analyses conducted using commercial codes, namely ANSYS (Stolarski et al., 2007) and DIANA (TNO DIANA, 2015), are also reported, where for masonry standard softening materials (e.g., smeared crack or total strain models) are assumed.

16.2 The nonuniform rational b-spline model: from geometry to limit analysis FRP-reinforced masonry vaults can be modeled in any free form 3D modeler using NURBS surfaces for both FRP strips and the underlying masonry structure, which can be adequately described by its mean surface. NURBS basis functions share many similarities with B-spline basis functions, to

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which they are strongly connected. B-spline basis functions are piecewise polynomials defined by the so-called knot vector Ξ 5 fξ1 ; ξ2 ; . . .; ξn1p11 g, composed by points in the parameters space ξ i A½0; 1, known as knots. In the expression of the knot vector, p denotes the degree of the polynomials whereas n is the overall number of base functions. The ith B-spline basis function, Ni;p , is obtained through the Cox
de Boor recursion equation (Piegl and Tiller, 1995). The NURBS base functions, Ri;p , are expressed as follows: Ni;p ðξÞwi Ri;p ðξÞ 5 P : n Ni;p ðξÞwi

ð16:1Þ

i51

where wi Aℝ are the weights. A NURBS surface of degree p in the u direction and q in the v direction is a parametric surface in the 3D Euclidean space defined as follows: Sðu; vÞ 5

n X m X

Ri;j ðu; vÞBi;j

ð16:2Þ

i50 j50

where fBij g form a bidirectional net of control points. A set of weights fwi;j g and two separate knot vectors in both the u and v directions must be defined. Many commercial free-form surface modelers, such as Rhinoceros (McNeel, 2008), utilize NURBS representation and its properties to generate and manipulate surfaces in the 3D space. Fig. 16.1A represents an example of a Rhino 3D model of a cross-vault reinforced with FRP strips at both the extrados and intrados of the four main arches; both FRP strips and the midsurface of the masonry structure can be modeled as NURBS surfaces. Typically, the easiest way to generate a NURBS mesh on a given surface is to define a subdivision of the 2D parameters space u 2 v, which follows from subdividing the knot vectors in both the u and v directions into equal intervals (see Fig. 16.1B). The resulting mesh is defined by isoparametric curves on the surface in the 3D Euclidean space. Each element of the mesh is a NURBS surface and its edges are branches of isoparametric curves belonging to the initial surface. More precisely, the counterimage of each element of the mesh is a rectangle Sij 5 ½ui ; ui11  3 ½vj ; vj11 Aℝ2 defined in the parameters space. Fig. 16.1C shows the structural mesh obtained within a MATLAB environment after exporting the original NURBS structure through the IGES standard and subdividing the original surface in a set of smaller NURBS elements to which the real vault thickness is assigned. Each NURBS element is imported with its own thickness, which is also geometrically taken into account in order to properly distinguish between intrados and extrados reinforcement (see Fig. 16.1C). In general, different meshes of the NURBS surface can be obtained for arbitrary partitions of the parameters space into quadrilateral or triangular domains. Particular attention must be addressed to multiple

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FIGURE 16.1 Generic double curvature FRP-reinforced masonry structure. (A) 3D NURBS model obtained with Rhinoceros; (B) parameters space subdivision; (C) corresponding subdivision of the vault midsurface and detail of the actual NURBS thickness with FRP applied at the extrados.

NURBS surfaces composed by Boolean operations between elementary NURBS patches, which is particularly frequent when dealing with geometries of typical masonry vaults. In that case, the counterimage of the surface in the parameter space is not a rectangle of ℝ2 anymore. The union of every element of the chosen mesh is equal to the original surface, no matter how coarse the mesh is.For each element of the mesh Ei, be the domain Ki its counter-image in the two-dimensional parameters space u
v. Ei , Ki .

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The area of the surface can be simply computed through the following relation: ðð ðð dS 5 :Su 3 Sv : du dv ð16:3Þ Ai 5 Ei

Ki

where Su and Sv are partial derivatives of the parametric surface Sðu; vÞ in the u and v directions. Analogously, the position of the centroid of each element (which is essential for the limit analysis discussed hereafter) may be computed with the following relation: ðð ðð 1 c5 x dS 5 Sðu; vÞ:Su 3 Sv : du dv ð16:4Þ Ai E i Ki An isoparametric approach can be adopted for the numerical computation of areas of curved surfaces and centroids. As depicted in Fig. 16.2, let K be a quadrilateral domain in the parameters space with straight boundary lines and vertices ðui ; vi Þ; i 5 1; 2; 3; 4 arranged in counterclockwise order. First, the quadrilateral domain K is transformed into the standard quadrilateral element Rst and then the Gaussian quadrature is applied, whose details and proof of convergence are discussed in Chiozzi et al. (2017a), where the reader is referred for further details.

16.2.1 Kinematic limit analysis Provided a NURBS 3D model of the FRP-reinforced vaulted surface, a NURBS mesh can be defined on the same surface and a discretization with a

FIGURE 16.2 (A) Linear mapping between K and Rst. (B) Masonry
masonry (left) and FRP
FRP interfaces and corresponding local frame of reference.

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set of rigid elements can be defined for both the masonry structure and the FRP reinforcement. Hence, starting from the geometrical properties of each element, an upper bound formulation of limit analysis can be easily formulated. In such a discretization, NURBS elements are considered rigid and thus internal dissipation is allowed only at the interfaces between contiguous elements. Indicating with NE the number of elements composing the NURBS mesh, the kinematics of each element is determined by the six (three translational and three rotational) generalized velocity components fuix ; uiy ; uiz ; Φix ; Φiy ; Φiz g of its center of mass Gi , expressed in a global reference system Oxyz. On the structure, dead loads F0 and live loads Γ, that is, the load multiplier, are acting. Internal dissipation is allowed along interfaces only. Interfaces (where all dissipation occurs) can be classified as follows: masonry
masonry, FRP
masonry, and FRP
FRP interfaces. Indicating with NITOT 5 NIM2M 1 NIM2F 1 NIF2F the total number of interfaces, the total internal dissipation power Dint is equal to the sum of the power dissipated along each interface and is also equal to the sum of the powers of live (1UΓ) and dead (F0 ) loads, indicated as PΓ and PF0 , respectively: Dint 5

Nl X

Piint 5 PΓ 1 PF0

ð16:5Þ

i51

The LP problem related to the kinematic formulation of limit analysis consists of an appropriate minimization of the load multiplier Γ under the action of suitable constraints. The vector of unknowns of the LP problem, X, contains the six generalized velocity components for each element and the sum of all plastic multipliers along each interface. Vertices belonging to element free edges, which do not constitute an element interface, can be subjected to external kinematic constraints, by imposing an assigned value for translational and/or rotational velocities at these points. For each of such vertex Vj , kinematic constraints can be expressed in terms of generalized velocities of the center of mass of the ith element they belong. For example, if only the translational velocities of a given vertex Vj , belonging to element i, are constrained to zero, the following relation holds as a geometric constraint:   uVj 5 ui 1 R xVj 2 xGi 5 0 ð16:6Þ  Vj Vj Vj T where uVj 5 ux ; uy ; uz are the three translational velocity components of the vertex Vj , ui 5 ½uix ; uiy ; uiz T are the three (unknown) translational velocity components of the center of mass of element i to which vertex Vj belongs, and R is the matrix collecting rotation rates of the element. In general, all linear geometric constraints can be rewritten in the following standard form: Aeq; geom X 5 beq; geom

ð16:7Þ

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where Aeq; geom is the matrix of geometric constraints and beq; geom is the corresponding vector of coefficients. Masonry
masonry and FRP
FRP interfaces are planar surfaces whose height in each point of their midline corresponds to the local structural thickness of the vault and of the reinforcement, respectively, whereas FRP
masonry interfaces are NURBS surfaces placed at the intrados and/or extrados of the vaulted surface. In order to enforce plastic compatibility along masonry
masonry interfaces and correctly evaluate dissipation power, intrados and extrados edges of each interface should be subdivided into an M assigned number ðNsd 1 1Þ of points Pi (three points are sufficient, provided they are not aligned, because the jump of velocities on interfaces is linear). With reference to Fig. 16.2B, on each point Pi of a masonry
masonry interface, a local reference system ðnM ; sM ; tM Þ has to be defined, where nM is the unit vector normal to the interface, sM is the tangential unit vector in the longitudinal direction, and tM is the tangential unit vector in the transversal direction. On each point Pi of each interface, which separates the two ele_ =@σÞ must hold, where ments E0 andEv, the compatibility equation Δu~ 5 λð@f σ 5 ½σnn ; σns ; σnt  is the stress vector acting on Pi in the three local refer_ is an unknown ence directions, f ðσÞ is a suitable strength function, and λ plastic multiplier vector. Δu~ is the representation in the local reference system of the quantity Δu in the global reference system, which is defined as Δu 5 u0 Pi 2 uvPi , where u0 Pi is the vector composed by the three translational velocity components of the point Pi seen as belonging to element E0 and ~ ~ is a uvPi . Δuis related to Δu~ through the relation Δu~ 5 RΔu, where R suitable rotation matrix whose rows are, respectively, the components of the three local vectors ðnM ; sM ; tM Þ expressed in the global reference system. The masonry strength domain for the interfaces f ðσÞ 5 0 can be obtained with a variety of rigorous or simplified homogenization procedures (e.g., in Cecchi et al., 2007; Milani and Cecchi, 2013; Milani and Taliercio, 2015; Milani, 2015b). Since the aim of this chapter is not to discuss the pros and cons of the different approaches available, the reader is referred to the Chapter 10 where such homogenization procedures are discussed. In any case, the resultant f ðσÞ 5 0 is typically furnished as a package of linear equations, which is the standard way a strength domain must be provided in linear programming. Assuming that the ith plane representing f ðσÞ 5 0 is given in the form Ai σnn 1 Bi σns 1 Ci σnt 5 1, the associated flow rule simplifies as follows: " pl #T N N pl N pl X i X i X i Δu~ 5 ð16:8Þ Ai λ_ Bi λ_ Ci λ_ i51

i51

i51

where λ_ is the ith plane plastic multiplier and N pl is the total number of linearization planes used. The previous constraint must hold for each point Pi of each interface. Since for each point of each interface a set of N pl unknown i

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plastic multipliers is defined, the total number of unknown plastic multipliers is equal to N pl ðNsd 1 1Þ2NI . On each interface i, covering the surface Si , the internal dissipated power is defined as the integral: ð M2M Pint;i 5 σUΔu~ dS ð16:9Þ Si

in the local reference system, where both σ and Δu~ have been previously defined. A Gauss quadrature method is adopted to perform numerically the previous integral. As far as FRP is concerned, plastic dissipation is allowed only along interfaces between adjoining elements, and is produced as an effect of longitudinal stresses acting in the direction of fibers. In order to enforce plastic compatibility along the FRP
FRP interfaces, the midline of each interface is F usually subdivided into an assigned number ðNsd 1 1Þ of points Pi , where a F F F local reference system ðn ; s ; t Þ is defined (Fig. 16.2B) with nF the unit vector normal to the interface (and parallel to the fiber direction); sF is the tangential unit vector in the longitudinal direction; and tF is the tangential unit vector in the transversal direction. Continuity of the velocity field must be enforced at interfaces adjoining the FRP elements along directions sF and tF only, while velocity jumps are allowed along direction nF , in order to simulate axial rupture. Different tension and compression limit stresses are usually imposed to account for the very low axial strength in compression of FRP due to buckling (a consequence of the very limited strips thickness), 1 respectively fFRP (equal to ffdd or ffdd;rid according to (CNR-DT200, 2013; see 2 the details reported hereafter) for crisis in tension and fFRP  0 for buckling in compression. In order to abide by kinematic admissibility, for every jump in velocity at an interface the following equality constraint, which particularizes the associated flow rule, must hold: T  T  Δu~ 5 Δu~n Δu~t Δu~s 5 λI2FRP1 ð16:10Þ 2λI2FRP2 0 0 i i and λI2FRP2 are plastic multipliers at point Pi on the where λI2FRP1 i i 1 2 FRP
FRP interface corresponding to fFRP and fFRP , respectively. Since for each point of each FRP
FRP interface a set of two unknown plastic multipliers is defined, the total number of unknown plastic multipliers for F FRP
FEP interfaces is equal to 2ðNsd 1 1ÞNIF2F . On the other hand, the power dissipated along the ith FRP
FRP interface may be evaluated as: ð ð   I2FRP1 I2FRP2 1 2 ~ PF2F 5 s σUΔ u dl 5 s λ 1 f λ f dl ð16:11Þ int;i FRP i FRP i Li

Li

where s is the thickness of the interface (i.e., of the FRP strip) and Li is the midline of the FRP
FRP interface considered.

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Adhesion between the reinforcement and the underlying material is a paramount aspect when considering the use of fiber-reinforced composites for retrofitting masonry structures. Many experimental works in the literature have proven that the delamination crisis is governed by failure in masonry (Ceroni et al., 2003), with FRP strips peeling a nonnegligible portion of the brickwork. Today, many codes exist to assess delamination of FRP strips from their support by using simple but sound mathematical relations. In order to keep things simple, in the following discussion we adopt the suggestions contained within the Italian technical norm (CNR-DT200, 2013), but adapt them to limit analysis environment. There, a conventional approach is used, consisting of adequately limiting the traction force acting within the FRP strip. If the bond length lb is greater than the optimal bond length le, the design tensile strength of a FRP strip ffdd is given by the relation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2UEFRP UΓ Fk ffdd 5 ð16:12Þ pffiffiffiffiffiffi γ fd γ M tFRP whereas in case lb # le ffdd is given by the following:   lb lb 22 ffdd;rid 5 ffdd le le

ð16:13Þ

In the previous relations, ffdd and ffdd,rid are the design bond strength and the reduced design bond strength, respectively; EFRP is the elasticity modulus of FRP; tFRP is the FRP thickness; γ fd is a safety factor equal to 1.20; γ M is the partial safety factor for masonry material, which in this case is assumed equal to 1.0;ffi lb is the bond length of FRP strips; and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi le 5 ðEFRP UtFRP Þ=2Ufmtm is the optimal bond length, equal to the smallest bond length capable of sustaining the maximum anchoring force. The quantity ΓFk is crucial, because it is the specific fracture energy of the FRP-reinforced masonry upon delamination. The Italian norm suggests for the FRP
masonry interface a bilinear (elastic followed by linear softening) τ b-slip constitutive law, which permits to indirectly estimate the shear limit stress fb to be employed for FRP
masonry interfaces, once that ultimate slip is known. From the results found in the literature, it could be inferred that a damaging material model would be a better choice when trying to assess the mechanical response of FRP-reinforced masonry structures (Luciano and Sacco, 1998; Marfia and Sacco, 2001). Such behavior, in principle, precludes the use of limit analysis, which is based on the infinite material ductility and perfect plasticity assumptions. Nevertheless, in agreement with the suggestions contained in CNR-DT200 (2013), limit analysis is still recommended to get a quick and reliable evaluation of the load-bearing capacity of a given

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masonry structure. In fact, the code (CNR-DT200, 2013) acknowledges that the effectiveness of any reinforcement is in the inhibition of the failure mechanism that would have deployed in the unreinforced case, once the designer has ruled out FRP delamination through a conventional approach (i.e., by suitably reducing design tensile strength of composite fibers). The subsequent change in the failure mechanism, according to the upper bound theorem, reveals an increase in the bearing capacity of the structure. Moreover, other authors have successfully investigated the capabilities of limit analysis for the assessment of FRP-reinforced masonry structures (see, e.g., Ascione et al., 2005; Caporale et al., 2006). In summary, the principal consequence of a given retrofitting intervention through the use of FRP strips is to preclude the activation of the collapse mechanism causing failure in the unreinforced conditions, with the consequent generation of a different failure mechanism and a greater internal dissipated power and a resulting higher collapse multiplier. Since “hand” calculations cannot be easily carried out for general curved masonry structures with arbitrary loading, an upper bound procedure is especially suited for a fast and computationally inexpensive evaluation of the load-bearing capacity of general FRP-reinforced masonry vaults. For FRP
masonry interfaces, an assigned number NPM2F of control points is selected with a local reference system ðsI ; tI ; nI Þ, where sI is the unit vector tangential to the surface in the direction of fibers, tI is the tangential unit vector in the transversal direction, and nI is the unit vector normal to the interface. The stress vector written in such a local frame of reference is σ 5 ðτ s ; τ t ; σn Þ. An FRP
masonry interface failure surface is linearized as M2F M2F Ak τ si 1 Bk τ ti 1 Ck σni 5 Dk , k 5 1; . . .; NPL (where NPL is the number of planes) and the associated flow rule is written as follows: 3 2 M2F N PL X M2F;k 7 6 Ak λ_ i 7 6 k51 7 6 2 3 7 6 M2F Δuis 7 6 NX 6 7 6 PL M2F;k 7 i ð16:14Þ ½ui  5 4 Δut 5 5 6 7 Bk λ_ i 7 6 i k51 7 6 Δun 7 6 M2F 7 6 NX 4 PL M2F;k 5 Ck λ_ i k51 M2F;k where i 5 A; B; C (nodes of the triangular element), whereas λ_ i is the kth plastic multiplier rate relative to the kth linearizing plane. The Italian design code for FRP reinforcement suggest specific σ
τ s
τ t failure surfaces for FRP
masonry interfaces, as depicted in Fig. 16.6B, in which fb is the interface shear strength and fmt describes the masonry tensile strength.

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An important body of experimental work clearly illustrated that strip delamination along curved structural surfaces always happens in the presence of nonnegligible normal forces between the support and the reinforcing element (see, e.g., Grande et al., 2008; Fedele and Milani, 2011; Grande and Milani, 2016; Bertolesi et al., 2018). Therefore, estimating the actual strength of the interfaces along the perpendicular direction is crucial. It may be of interest to observe that interface strength with respect to normal stresses is here considered equal for both tension and compression. In a more sophisticated model, different limiting strength for tensile and compressive behavior should be assumed, fmt being closely connected to the tensile and compression strength of bricks. It should be noted that in the present model, any nonlinear failure surface for the FRP
masonry bond could be included without any particular theoretical complication. On each FRP
masonry Ðinterface i of area Si , the internal dissipated M2M power is defined as Pint;i 5 Si σUΔu~ dS in the local reference system, where both σ and Δu~ have been defined previously. Typically, such integral is evaluated numerically through a Gauss quadrature method. Two additional classic constraints to be considered are (1) the nonnegativity of all plastic multipliers, that is, λ_ ij $ 0, (2) and the so-called normalization condition that requires that the external power dissipated by the live load 1UΓ be equal to one, that is, PΓ51 5 1, which consequently defines the TOT NP I collapse multiplier as Γ 5 Piint 2 PF0 . i51

Taking into consideration all previously discussed inequality and equality constraints and applying the kinematic theorem of limit analysis, the related LP problem allows us to establish the collapse multiplier as the minimum of the internal power dissipated minus the power expended by loads independent from the failure multiplier: ( ) NI X min Piint 2 PF0 ð16:15Þ i51

under geometric constraints, compatibility constraints, nonnegativity of plastic multipliers, and normality condition. The unknowns of the LP problem are the 6UNE generalized velocity components of the center of mass of each element and the total number of plastic multipliers at each point of each interface.

16.3 Genetic algorithm mesh adjustment The only dissipation allowed in the model is that along interfaces; as a consequence, it is necessary to introduce an algorithm that allows us to adjust the mesh in order to find the minimum collapse multiplier among all possible configurations and therefore to determine the actual collapse mechanism. A GA is adopted with this aim.

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A NURBS mesh of a vaulted surface is determined by a given number Npar of real parameters p1 ; p2 ; . . .; pNpar that depend on the type of collapse mechanism that must be detected. A given NURBS mesh is regarded as an individual and each individual is written as an array with 1 3 Npar elements as individual 5 ½p1 ; p2 ; . . .; pNpar . The cost of each individual is evaluated using a cost function f at the parameters p1 ; p2 ; . . .; pNpar . The function f is defined as a function that outputs the collapse load multiplier λc for every assigned individual (i.e., an assigned mesh on the surface) through the implementation of the limit analysis procedure previously described. To initialize the GA, an initial population of Nipop individuals is defined. The whole population is represented by a matrix in which each row is a 1 3 Npar array (i.e., the individual) of continuous parameters. In order to establish chromosomes belonging to the initial population that are suited to survive and produce offspring for the next generation, the cost of theNipop individuals is ranked from the lowest to the highest. The best Npop individuals for the next algorithmic iteration are retained and the rest are left to die. This process is called natural selection, and from this point on, the size of the population at each generation is Npop . Other and more sophisticated selection operators have been introduced in the literature (Goldberg, 1989; Kwon et al., 2003) as, for example, proportional and tournament selection operators. Then, an equal number of mothers and fathers is selected within the Npop individuals, which pair in some random fashion. There are various reasonable ways to pair individuals. In this chapter, a weighted cost selection with assigned probabilities is used (Haupt and Haupt, 2004). Each pair of individuals generates two offspring containing traits belonging to each of the parents. Mating is carried out by choosing an assigned number of crossover points (at least one) on the chromosome. The parameters between these points are simply swapped between the parents. In this chapter, a crossover operator with multiple points is employed. The crossover points ki 5 ½1; 2; . . .; c 2 1 are selected in a random way on each pair of two individuals, represented by c chromosomes. Moreover, care must be taken in order to prevent the GA from converging too fast into one given region of the cost surface. Indeed, this may be not good if the problem being modeled has several local minima, in which the solution may get trapped. In order to circumvent issues related to exceedingly fast convergence, the algorithm is forced to test other regions of the cost surface by adding random changes, known as mutations, for a given share of parameters. A classic mutation operator is applied to all Npop individuals at each generation. For each individual pi the mutation operator acts in a stochastic way on all the chromosomes of the individual subject to mutation (i.e., randomly changing one of the chromosomes in the individual involved in the process of generating offsprings).

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The algorithm described can be improved by adding a zooming with elitist strategy (see, e.g., Milani and Milani, 2008) with the aim of considerably enhancing both algorithmic efficiency and robustness. In particular, the zooming approach provides for a subdivision of the initial population into two sets of individuals x 5 fxi :i 5 1; . . .; Nelit g and y 5 x 2 x 5 fyi :i 5 1; . . .; Npop 2 Nelit g that, for each iteration, collects individuals with the highest fitness into an “elite” subpopulation. The dimension of this subpopulation is defined by the user and is denoted by Nelit . Subsequently, to each individual in the set x, only a high probability mutation is assigned (i.e., not a crossover) in order to improve individual fitness. From a practical viewpoint, zooming must be user defined through the assignment of the commonly named zooming percentage z% , which describes the ratio (in percent) between the sizes Npop and Nelit of the initial population and the subgroup x, respectively. Even though zooming percentage is assumed constant (equal to 5%), it can still be suitably reduced when passing from an iteration to the successive using an exponential reduction scheme. A logic tree for the algorithm adopted is depicted in Fig. 16.7.

16.4 Numerical simulations in limit analysis In this section, two meaningful examples of technical relevance are discussed. The results from GA
NURBS based limit analysis are compared with both experimental data obtained from the literature (when available) and alternative numerical procedures. Each example is analyzed considering both the unreinforced configuration and the presence of FRP reinforcement to evaluate the capability of the proposed model to accurately detect both the ultimate load-bearing capacity and the different failure mechanisms exhibited by reinforced and unreinforced vaults. The first set of examples is represented by a straight and a skew unreinforced parabolic arch (tested in Vermeltfoort, 2001). For these two examples, experimentation was carried out only in the unreinforced configuration, since the original campaign did not contemplate testing on FRP-reinforced arches. The straight arch is a rectangular barrel vault with a 3.00 m clear span, a 2.50 m inner radius, a 1.25 m width, and a 0.50 m sagitta, as depicted in Fig. 16.3A, which shows geometry and loading conditions. The second example (Fig. 16.3B) has the same geometry, but supports are skewed with an offset perpendicular to arch axis of 1.25 m. Skewness conduces to a highly tridimensional behavior, which does not permit to employ a 1D model. Both arches have a depth equal to 100 mm. Bricks have dimensions 200 3 100 3 52 mm3, with a 12 mm joint thickness. As stated in Vermeltfoort (2001), the compressive strength of the bricks has been measured through compression tests and is equal to 27 N/mm2, while a compressive strength of 2.5 N/mm2 was found for mortar. As depicted in Fig. 16.8, during experiments, the only point load that was increased until collapse was

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FIGURE 16.3 Straight (A) and skew (B) parabolic arches tested by Vermeltfoort (2001); geometry and loading condition.

the second from the left, while the remaining loads were kept at a value of 5.0 kN throughout the test. When dealing with the unreinforced straight parabolic arch, the NURBS mesh of the vaulted surface is generated by four moving interfaces in the parameters space as illustrated in Fig. 16.4A. The arch classically collapses for the formation of four flexural hinges; considering the arch geometry and the disposition of dead and live loads, it is expected that the interfaces will remain orthogonal to the longitudinal direction of the structure, which justifies the choice of the four aforementioned parameters to describe the adaptive NURBS mesh (the position of each interface is indeed governed by one parameter only). On each interface Nsd is set equal to one to further speed up computations. For GA parameters, an initial population of 10 individuals is assued, each individual being a four-element vector. After optimization, a collapse load multiplier λ 5 43:35 is obtained. Fig. 16.4C shows the 3D NURBS-deformed shape at collapse after the optimization of the position of the hinges. The four-hinge collapse mechanism is comparable to the one observed by Vermeltfoort (2001) during experiments. As can be noted from Fig. 16.4E, where the best and mean fitness at successive iterations are depicted, the algorithm has fast convergence to the optimal solution and the final best fitness value is obtained after only few generations. In Fig. 16.6A, a comparison among load
displacement curves obtained with the commercial FE code DIANA (TNO DIANA, 2015), experimental results by Vermeltfoort (2001), and the homogenized limit analysis presented by Milani et al. (2009) is reported. The collapse load obtained using a simple manual 1D approach described in Focacci (2008) is also reported. As can be

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FIGURE 16.4 (A and B) 3D NURBS model of the unreinforced straight (left) and skew (right) parabolic barrel vault by Vermeltfoort (2001) generated with Rhinoceros; (C and D) collapse mechanisms from kinematic limit analysis with the proposed GA-NURBS (left: straight arch; right: skew arch); (E and F) convergence of the genetic algorithm towards the optimal solution in terms of best fitness and mean value (left: straight arch; right: skew arch).

observed, very good agreement between the results obtained through the present GA
NURBS approach and other more expensive methods is achieved. As a matter of fact, various models essentially produce identical failure loads, as well as very similar collapse mechanisms.

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FIGURE 16.5 (A and B) Collapse mechanisms from kinematic limit analysis with the proposed GA-NURBS in presence of FRP (left: straight arch; right: skew arch); (C and D) Convergence of the genetic algorithm towards the optimal solution in terms of best fitness and mean value (left: straight arch; right: skew arch).

To prevent the formation of the hinges observed in the unreinforced case, we can imagine reinforcing the arch with two FRP strips 100 mm wide at the extrados, as in Fig. 16.5A. A bond strength fb equal to 0.3 MPa is assumed in the simulation in agreement with CNR-DT 200 indications (CNR-DT200, 2013). The NURBS mesh of the vault, governing parameters, number of subdivisions of the interfaces, and population used in the GA remain the same used in the unreinforced case. A collapse load multiplier λ 5 57:02 is obtained. Fig. 16.5A shows the 3D NURBS deformed shape at collapse. Again, a fourhinge collapse mechanism is observed, but it appears clear that the reinforcement helps to counteract the creation of the hinges causing failure in the unreinforced configuration. Furthermore, strip delamination is evident at the hinges. Here, an important issue should be noted, namely that in the present formulation all elements, including FRP ones, are treated as rigid-infinitely resistant. This means that, given the usually big relative size of each FRP element, a huge power dissipation is required for delamination taking place away from the element edges. Thus, FRP reinforcement fails exactly along the element edges by reaching the critical tensile strength, an outcome in agreement with the real behavior of FRP. The numerical critical tensile strength already takes into account delamination in a conventional way by suitably reducing the actual tensile strength of FRP elements as suggested in CNR-DT200 (2013).

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As can be appreciated in both Figs. 16.4E and F, and 16.5C and D, the metaheuristic GA algorithm exhibits an impressively fast convergence toward the optimal solution. As a matter of fact, the converged final best fitness is obtained after only a very few generations. By comparing deformed configurations in the absence or presence of reinforcement (Figs. 16.4E and 16.5C, respectively), it is clear that the application of FRP strips produces a quite visible change in the collapse mechanism, which is linked to an increase in the collapse load multiplier of the reinforced structure. It is also worth noting that the NURBS approach is capable of distinguishing automatically between intrados and extrados reinforcement, because NURBS elements are implemented in the code with their actual geometric thickness. In order to verify such features, the same limit analysis problem is rerun in the presence of a reinforcement applied at both intrados and extrados. In this latter case, the collapse load multiplier raises to λ 5 80:08 (with an essentially unchanged collapse mechanism), demonstrating the effectiveness of the approach adopted. Similar to the unreinforced case (Fig. 16.6C) a comparison among the obtained results and (1) load
displacement curves obtained using the

FIGURE 16.6 Comparison among the results obtained from the proposed GA-NURBS approach and experimental data contained in Vermeltfoort (2001), FEM nonlinear simulations (DIANA) and homogenized limit analysis proposed in Milani et al. (2009) for the straight ((A) unreinforced; (C) reinforced) and skew ((B) unreinforced; (D) reinforced) parabolic barrel vaults.

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commercial FE code DIANA (TNO DIANA, 2015) and (2) collapse loads furnished by both the homogenized limit analysis presented in Milani et al. (2009) and the simplified “at hand” calculation by Focacci (2008) is given. The second benchmark is the analysis, in the absence and presence of reinforcement, of the same parabolic arch but in a skew configuration. The skew shape forces the arch to fail nonsymmetrically, displaying a mixed interaction of bending and torsional motions following yield lines. The NURBS mesh of the vaulted surface is generated by four moving interfaces in the parameters space as depicted in Fig. 16.4B. Differently from the straight configuration, the moving interfaces can also rotate, adding to the problem four additional parameters (rotations) and resulting in a limit analysis problem governed by a total of eight position parameters. On each interface, a number of subdivisions equal to Nsd 5 6 is adopted. An initial population in the GA of 10 individuals is chosen, with each individual being an eight-element vector. In the unreinforced case, a collapse load multiplier λ 5 27:87 is obtained (Fig. 16.4F), again in good agreement with experimental evidences (26 kN), whereas for the reinforced arch λ 5 43:91 (Fig. 16.5D) is obtained. Again, the convergence exhibited by the GA algorithm seems very good in both cases. In Fig. 16.6B and D, a comparison among GA-NURBS collapse load predictions, load
displacement curves obtained with a heterogeneous FE nonlinear approach (TNO DIANA, 2015), experimental results—where available—by Vermeltfoort (2001), and the homogenized limit analysis presented by Milani et al. (2009) in the absence (Fig. 16.6B) and presence (Fig. 16.6D) of reinforcement (only extrados or extrados 1 intrados) is reported, where the aforementioned good agreement is noticeable. As can be observed, in the presence of a reinforcement glued at both intrados and extrados the load-carrying capacity increases up to λ 5 54:12 kN (see Fig. 16.6D). In general, it should be noted that the proposed GA-NURBS limit analysis approach tends to slightly overestimate the bearing capacity of the masonry structure with respect to both FE simulations and experimental evidences. This fact is a clear consequence of adopting an upper bound formulation. The second structural example considered is a hemispherical dome with a 2.20 m inner radius 12 cm thick, experimentally tested in Foraboschi (2006). The dome is loaded up to failure through a concentrated vertical force as depicted in Fig. 16.7. Due to the axisymmetric configuration, the NURBS mesh of the dome, whose 3D model is reported in Fig. 16.7B, is generated by only one moving interface in the parameters space. This interface traces a parallel of the dome in the Euclidean space, whereas other eight fixed interfaces in the orthogonal direction define the number of meridians subdividing the dome. Nsd is set again equal to 6 and the initial population of the GA is constituted by 10 individuals. A collapse load multiplier λ 5 52:20 is obtained in the

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FIGURE 16.7 Hemispherical dome. (A) Cross-section geometry and loading condition (top) and FRP disposition (bottom). (B) NURBS parameters space and interface counterimage with 3D NURBS model (unreinforced case) generated with Rhinoceros.

unreinforced case (see Fig. 16.8A and B). As can be observed, the collapse mechanism is constituted by one annular bending hinge and a detachment along meridians, basically caused by nonzero annular membrane forces. The fast convergence at the final best fitness is again worth noting. In the reinforced case (Fig. 16.8C and D) collapse load increases up to λ 5 87:39 kN, clearly demonstrating the beneficial hooping effect of the three annular FRP strips introduced as reinforcement. In the failure mechanism found (Fig. 16.8C), it is clear how hooping strips partially preclude the creation of meridian cracks, forcing cracks to localize more around the application area of the live load, with a perceivable out-of-plane sliding of the top portion. Analogous with the parabolic barrel vault example, in Fig. 16.9 a comparison among load
displacement curves obtained with heterogeneous FE nonlinear computations (TNO DIANA, 2015), experimental results by Foraboschi (2006)—where available—and the homogenized limit analysis presented by Milani et al. (2009) is reported. As can be observed, in both the unreinforced and reinforced case, good prediction of collapse loads is obtained with the GA-NURBS approach, with a required computational time needed to optimize the coarse NURBS mesh requiring only a fraction of the computational time needed by commercially available inelastic FEM solvers.

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FIGURE 16.8 Hemispherical dome. (A) Unreinforced case, deformed shape at collapse after mesh optimization. (B) Convergence of the genetic algorithm towards the optimal solution in terms of best fitness and mean value. (C) FRP-reinforced case, deformed shape at collapse after mesh optimization. (D) Convergence of the genetic algorithm towards the optimal solution in terms of best fitness and mean value.

FIGURE 16.9 Comparison among the results obtained from the proposed GA-NURBS approach and experimental data contained in Foraboschi (2006), FEM nonlinear simulations (DIANA) and homogenized limit analysis proposed in Milani et al. (2009) for the unreinforced (A) and reinforced (B) case.

16.5 Nonlinear modeling Generally speaking, any limit analysis approach for FRP-reinforced masonry (such as the previously presented one), exhibits three main obstacles: (1) formally, its application is only valid for perfectly plastic materials, a condition

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that is rigorously not applicable to brittle phenomena; (2) it gives no indication about the behavior of the structure under serviceability conditions; and (3) it does not provide any information on displacements at failure. For these reasons, a full nonlinear analysis, which includes successive stages, from the absence of load except self-weight, passing through the behavior under service conditions and ending with the nonlinear behavior up to collapse, is ideally the most comprehensive form of structural analysis for static loads (Grande et al., 2008) that can be conceived. Currently, many commercial programs use general-purpose nonlinear analyses for fragile materials that are adaptable to masonry structures (e.g., Smith, 2016; Stolarski et al., 2007) and are easily accessible at a professional level, but refined models taking into account the complex interaction between FRP and masonry substrate are still in development. There are obviously some open issues to tackle related to convergence difficulties usually arising in the presence of strongly nonlinear problems and softening materials. In addition, the utilization of such software is usually not straightforward since, in general, the possibilities of modern FE codes far exceed engineers’ knowledge. Since earthquakes are a major source of damage and a recurrent extreme event, nonlinear analyses are becoming quite popular for existing masonry structures, and the interpretation of the effect induced by external FRP retrofitting in the nonlinear range is quickly receiving growing attention. In order to circumvent some typical drawbacks of standard FEM in the softening range, in Milani and Lourenc¸o (2013a,b) the nonlinear problem of masonry reinforced with FRP was recently reformulated in a nonlinear programming framework using a sequential quadratic programming scheme capable of following the nonlinear pushover curve in the softening range. From a numerical point of view, the choice of using rigid blocks with deformations lumped at interfaces in spite of standard FEs is essentially inspired by the need for variables reduction, but also reproduces the actual deformations pattern in the nonlinear field at least for masonry with sufficiently regular texture, where damage usually propagates on small fracture lines, zigzagging along joints between adjacent bricks. This approach potentially allows us to perform numerical simulations on entire medium/largescale structures in a fraction of the time required by standard FE methods. The utilization of discrete elements has a long tradition in the mechanics of structures and goes back to the pioneering work proposed by Kawai (1978). The utilization of rigid elements with a very reduced number of degrees of freedom, and the efficiency of the procedure tested for unreinforced panels in two-way bending (Casolo and Milani, 2010), suggested a further improvement of the nonlinear model, relying into its extension to in-plane loaded panels (Milani, 2011a) and its successive generalization to curved masonry shells (Milani and Tralli, 2012). Several additional complications arose in this latter case connected to the increase of variables to be handled per interface (modeled using five nonlinear springs); the introduction of a

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FIGURE 16.10 Nonlinear structural analysis of FRP-reinforced masonry with rigid elements and interfaces (wedges for masonry and triangles for FRP).

simplified dependence between out-of-plane behavior and membrane internal actions; and the utilization of more efficient large-scale quadratic programming routines to handle more realistic engineering problems and hence many more elements in the FE discretization. The final improvement was finally presented in Milani and Lourenc¸o (2013a,b), with the implementation of external reinforcements such as FRP strips and tendons and steel tie rods. This chapter reviews this latter FE model where (1) external reinforcement is introduced at a structural level and modeled using the utilization of plate and shell elements with deformation allowed at the linear interfaces between adjoining elements and (2) a brittle 2D constitutive behavior between FRP and support is introduced that fully complies with current recommendations (e.g., CNR-DT200, 2013). In particular, the following subsections quickly review the essential features of the nonlinear homogenization model, namely the elements used to represent at a structural level masonry, FRP, FRP
masonry interfaces, and the general quadratic programming scheme utilized to tackle the nonlinear elasto-plastic problem at each load step (Fig. 16.10).

16.5.1 Masonry wedge elements At a structural level, for masonry, six-noded wedge elements interconnected by homogenized four-noded planar interfaces are utilized. The motion of a generic element E (see Fig. 16.11A) is described as a function of its centroid (C E ) displacement vector uE (components uExx , uEyy and uEzz ) and of its rotation vector ΦE (components ΦExx , ΦEyy and ΦEzz ) around the centroid. When two contiguous elements M and N are considered (see Fig. 16.11B and C) the displacement of a generic point P in a position ξAΓ12 belonging,

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FIGURE 16.11 (A) Rigid infinitely resistant six-noded wedge element used to model masonry at a structural element; (B) Γ12 interface between contiguous elements; (C) jump of displacement on a point P on the M
N interface and corresponding state of stress.

respectively, to M and N (where Γ12 indicates the common interface between the two elements) is: uM ðPÞ 5 uM 1 MðΦM ÞðP 2 CM Þ

ð16:16Þ uN ðPÞ 5 uN 1 MðΦN ÞðP 2 C N Þ 2 3 0 2Φzz Φyy 0 2Φxx 5 where MðΦÞ 5 4 Φzz 2Φyy Φxx 0 A jump of displacements ½UðPÞ between bricks M and N in a point P is therefore expressed by:

UðPÞ 5 uM ðPÞ 2 uN ðPÞ 5 uM 2 uN 1 MðΦM ÞðP 2 C M Þ 2 MðΦN ÞðP 2 CN Þ ð16:17Þ Having defined a local frame of reference ξ1
ξ2
ξ 3 for the interface between N and M elements (vertices corresponding to nodes P1 , P2 , P4 , and P5 , Fig. 16.11B and C), we assume that it is characterized by two axes (e1
e2 ) laying on the interface plane and mutually orthogonal, while the third perpendicular axis to the interface is e3 . Thus, unitary vectors e1
e2
e3 may be expressed in the global coordinate system as e1 5 e2 3 e3 , e2 5 ðP2 2 P1 Þ=:P2 2 P1 : and e3 5 e~ 1 3 e2 with e~ 1 5 ðP4 2 P1 Þ=:P4 2 P1 :. The rotation matrix Re , with respect to the global coordinate system jump of displacements, may be written in the local system as: ~ ½UðPÞ 5 Re ½UðPÞ

ð16:18Þ

where the superscript ~ indicates the quantities evaluated in the local system. From Eq. (16.18), it is possible to evaluate the work W expended on Γ12 interface as follows: ð ð   ~ ~ ~ W 5 σM ðPÞUUM ðPÞ 1 σN ðPÞUUN ðPÞ dS 5 σM ðPÞU½UðPÞ dS ð16:19Þ I

I

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FIGURE 16.12 (A) FRP/masonry interface delamination law adopted (with its linear piecewise approximation useful to use quadratic programming); (B) FRP triangular element (left) and A
B interface between two contiguous M
N triangular FRP elements (right), with corresponding local frame of reference. (C) FRP/masonry interfaces. Discretization of interfaces with triangular elements interacting with FRP triangles and masonry wedges.

 T where σM ðPÞ 5 τ 13 ðPÞ τ 23 ðPÞ σ33 ðPÞ is the stress vector acting at P on element M, with σN ðPÞ 5 2 σM ðPÞ. To model FRP, rigid infinitely resistant triangular elements are adopted as in Fig. 16.12. Being rigid, elastic and inelastic deformation is hence allowed only at the interfaces between contiguous elements (linear FRP
FRP interfaces or triangular FRP
masonry interfaces).

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Considering a (k)th FRP strip with direction sðkÞ and two contiguous FRP elements M and N, with centroid displacements and rotations defined as h iT  T  T M uM uM ΦM , uN 5 uNxx uNyy uNzz , ΦM 5 ΦM uM 5 uM xx yy zz xx Φyy zz h iT and ΦN 5 ΦNxx ΦNyy ΦNzz , the jump of displacements on the common M and N interface (I-FRP) is linear; therefore, its evaluation is only necessary on the interface extremes A and B, calculated as the difference between displacements of nodes 1
3 and 2
4, respectively. A jump of displacements is evaluated (in the local coordinate system) as: ½uA  5 TðM; NÞ½uM 2 uN 1 RM ðA 2 CM Þ 2 RN ðA 2 C N Þ ð16:20Þ     T T where ½uA  5 ΔuAs ΔuAr ΔuAt 5 u1s 2u2s u1r 2u2r u1t 2u2t is the ðkÞ ðkÞ ðkÞ jump of displacement, s
t
r is a local frame of reference, and TðM; NÞ is a rotation matrix between the local and global frame of reference (see also Fig. 16.12 for symbol meanings and Milani and Lourenc¸o, 2013a,b for further details on the model). Analogous considerations can be repeated for the other extreme of the interface. As already discussed for the limit analysis model, elastic and inelastic deformation is supposed to occur at the interfaces only, due to stresses acting both parallel and perpendicular to the sðkÞ fiber direction. Low compressive stresses induce buckling of the strips, due to the FRP negligible thickness. In order to take into account this effect (at least in an approximate way), differ1 ent limit stresses are assumed in tension and compression, namely fFRP (assumed equal to ffdd or ffdd;rid in agreement with CNR-DT200, 2013 for 2 tensile failure) and fFRP  0 for compression buckling, respectively. Correct softening modeling of the FRP
masonry interface is paramount for correct interpretation of delamination of the FRP from the support, and the present nonlinear analysis approach can account for the actual τ b (interface shear stress)
slip law proposed by the Italian norm (Fig. 16.12). Such a relationship, in agreement with the previous limit analysis model, permits an indirect evaluation of shear peak stress fb to use for masonry
FRP interface elements (and thus avoiding discretization of FRP strips using truss elements with limited strength ffdd), once the ultimate slip (usually fixed at 0.3 mm) is known (area under the τ b -slip constitutive law of Fig. 16.12 is ΓFd ). In addition, the nonlinear model is able to account for the postpeak linear softening behavior, which is approximated in the framework of the sequential quadratic programming approach hereafter reviewed, using a linear piecewiseconstant discontinuous function, as depicted in Fig. 16.12A. For a triangular FRP
masonry interface I F2M between elements F (FRP) and  T  T uM uM M (masonry) (Fig. 16.12) uF 5 uFxx uFyy uFzz and uM 5 uM xx yy zz indicate F and M centroid displacements, respectively, whereas h iT h iT ΦM ΦM and ΦM 5 ΦM F and M are rotation ΦF 5 ΦFxx ΦFyy ΦFzz xx yy zz

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vectors. Jump of displacements on the common I F2M interface is linear and may be evaluated on nodes A, B, and C of the interface (Fig. 16.12) as the difference between displacements of nodes 1
4 and 2
5 and 3
6, respectively. The procedure is analogous to those followed for masonry
masonry and FRP
masonry interfaces and is not repeated here for the sake of conciseness. Interested readers are referred to Milani and Lourenc¸o (2013a,b) for further details. Following the relatively long tradition of the distinct element method and the rigid body and spring modeling (Casolo and Milani, 2010, 2013; Milani et al., 2012; Bertolesi et al., 2017), rigid elements are assumed interconnected on interfaces using nonlinear springs. When dealing with masonry material, three displacement and two rotational nonlinear springs are utilized, as schematically shown in Fig. 16.13. The third rotational spring acting along an axis parallel to the surface is assumed rigid and infinitely resistant. For FRP
masonry interfaces, three displacement springs per node are assumed. A step-by-step solution technique based on quadratic programming is adopted, whose computational details are available in Milani and Lourenc¸o (2013a,b) for interested readers. Here, it is worth noting only that the utilization of quadratic programming routines in sequence requires the approximation of the nonlinear force
displacement (or stress
strain) relationships assumed for the springs using linear piecewise constant discontinuous functions, as depicted in Fig. 16.12A. In addition, it is worth mentioning that the utilization of springs results in a partially decoupled modeling for in- and out-of-plane actions. To properly take into account some distinctive aspect of masonry behavior in flexion

FIGURE 16.13 Schematic representation of displacement springs acting on masonry and FRP
masonry interfaces.

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(dependence of the flexural behavior by in-plane compression), but limiting to a great extent the number of optimization variables involved in the quadratic programming scheme, a simplified recursive procedure where the inplane actions is assumed constant in the iteration substep is adopted. The entire algorithm is quite straightforward, but the details are omitted here for the sake of brevity. Again, interested readers are referred to Milani and Lourenc¸o (2013a,b). Within each iteration, an elastic-perfectly plastic approximation for each spring is utilized, meaning that 10 plastic multipliers (two for each spring, λ1 and λ2 , corresponding to positive or negative kinematic variables) for each interface are needed. Conversely, for FRP
masonry interfaces a total of 18 plastic multipliers are needed. Within the discretization adopted, the step-by-step nonlinear problem may be formulated as follows: 8 <

8 <1   ðλ1 2λ2 ÞT Kep ðλ1 2 λ2 Þ 1 UTel Kel Uel 2 FT wsubject to:λ1 $ 0 and min : :2

λ2 $ 0 ð16:21Þ

Assuming that the structural model has nel elements, symbols in Eq. (16.21) have the following meaning: 1. Kel is a 6nel 3 6nel assembled diagonal matrix collecting elastic stiffness of each interface (spring). It is worth remembering that elastic stiffness values are evaluated at the mesoscale, as discussed in the previous section. Since masonry is assumed at a structural level already homogenized and hence anisotropic, both in the elastic and inelastic range, they depend on the interface orientation angle. 2. λ1 and λ2 are two vectors of plastic multipliers, collecting plastic multipliers of each nonlinear spring present in the model. 3. Kep is the assembled diagonal matrix of hardening moduli of the interfaces. A small but nonzero hardening has to be introduced in order to avoid lack of convergence of the quadratic programming algorithm. 4. Uel is a 6nel vector collecting the displacements and rotations of the elements. 5. F is a 6nel vector of external loads (forces and moments) applied on element centroids. Typically, the independent variable vector is represented by element displacements Uel and plastic multiplier vectors λ1 and λ2 . Eq. (16.21) is solved at increasing values of the external load vector F and, at each external load value, the initial trust independent variable vector is the solution at the previous step.

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As usually done in a nonlinear structural analysis, quadratic programming problem Eq. (16.21) is solved in terms of displacement and plastic multipliers step increments. The initial trust independent variable vector is always represented by the solution at the previous step. A quite articulated step-by-step numerical procedure is implemented to solve Eq. (16.21) using the recursive (sequential) call of an efficient quadratic programming routine, whose details are available in Milani and Lourenc¸o (2013a,b) and Milani and Tralli (2012).

16.5.2 Nonlinear numerical simulations: deep beam Three masonry panels with and without CFRP strips strengthening and experimentally tested in Milani et al. (2006b) denoted as PAN, PAN-A1, and PAN-A2, are examined here to benchmark the nonlinear homogenization approach previously reviewed for in-plane actions (see Fig. 16.14). All panels, built with 1/4 of common solid clay Italian bricks (dimensions 62.5 mm 3 30 mm 3 14 mm), have dimensions 290 mm 3 270 mm (base3 height). PAN-A is the nonstrengthened wall, whereas PAN-A1 and PANA2 are specimens strengthened with a single horizontal strip in PAN-A1 and two symmetrical diagonal strips in PAN-A2. For these panels, several results are available, namely experimental data, nonlinear simulations with a commercial code, and homogenized limit analysis collapse loads (see Grande et al., 2008). The experimental tests were performed statically increasing the

FIGURE 16.14 Masonry deep beam. Geometry, loading condition, and FE discretization adopted for the nonlinear homogenization numerical analyses.

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vertical external load applied at the top edge. The obtained results in terms of force
displacement diagrams (i.e., vertical load applied versus displacement of the steel plate that transfers the load to the panel) show key aspects induced by the CFRP strengthening on the global response of the panels. Furthermore, examination of the crack paths during and after the tests shows important information on the effectiveness of a correct numerical modeling proposed and on the contribution of the strengthening. It is worth noting that (see Fig. 16.14) all series were placed on steel plates of length Ls equal to 40 mm, disposed at the lower-edge extremes and positioned on steel rollers allowing rotation of the supports. The rotation of the lower-edge extremes has a minor effect on the numerical results (Grande et al., 2008), and is not considered here for the sake of simplicity. Mechanical properties assumed for the constituent materials (brick and mortar) are—where possible—in agreement with experimental data available. The homogenization model adopted in Milani (2011a) and in Milani and Tralli (2011) to deduce homogenized stress
strain relationships for masonry to use at the macroscale is a simplified one (see also Milani and Bertolesi, 2017b), where the elementary cell is discretized with a coarse mesh constituted by elastic triangles (bricks) and nonlinear interfaces (mortar joints). For mortar, a frictional-cohesive behavior with limited tensile strength and linearized compressive cap (exhibiting hardening and softening) in agreement with the elasto-plastic model proposed in Lourenc¸o and Rots (1997) is adopted. The reader interested in the implementation details and the numerical parameters used is referred to Milani and Lourenc¸o (2013a,b). Experimental load
displacement curves for the three series of panels analyzed here (see Fig. 16.15) show that the introduction either of a horizontal strengthening (PAN-A1) or a double diagonal strengthening (PAN-A2) results in a considerable increase of the ultimate load and the ductility. In particular, in Fig. 16.15, (1) the force
displacement curves of the point of application of the external load (center of the steel plate) from the nonlinear homogenization approach previously reviewed, (2) the ultimate load from an upper bound FE limit analysis proposed in Milani (2011b), (3) the experimental force
displacement curves are reported for all the panels, and (4) simulations performed with the commercial code DIANA (TNO DIANA, 2015), where an orthotropic elasto-plastic with softening macroscopic model is adopted for masonry, are also represented to further assess numerical results. Full details of the latter model may be found in Grande et al. (2008). For the unstrengthened panel (PAN-A), it is interesting to note that the results obtained using the two-step approach here presented are, near the peak point, almost identical to the experimental data, furnishing also a strength value in very good agreement with the DIANA simulations. Also, the initial stiffness and the postpeak behavior are reproduced very well.

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FIGURE 16.15 Masonry deep beam. Left: comparison among load
displacement curves or collapse loads provided by experimentation, limit analysis, and nonlinear FE codes (commercial and nonlinear homogenization software). Center: deformed shapes at peak. Right: degraded interfaces patch for shear stress (from 0—no degradation—to 1—full degradation).

For the strengthened panel PAN-A1, the present model exhibits a force
displacement curve in good agreement with both experimental data and commercial code DIANA simulations, also in the postpeak range. The results obtained for PAN-A2 are again very near to experimental ones, both in terms of peak-strength and postpeak behavior. The acceptable differences between the homogenization nonlinear model and DIANA may be explained by remembering that within DIANA the strengthening is modeled using truss elements perfectly bonded to the masonry surface, where delamination is accounted for limiting tensile strength to ffdd or ffdd,rid near the anchorage zone. In Fig. 16.15 deformed shapes at peaks obtained for PAN-A, PAN-A1, and PAN-A2 series are also represented. As FE simulations show, in PAN-A1 series the horizontal strip acts as a tie. Even though the two-strut model of the unstrengthened case remains essentially unchanged, both the compressed sections increase as well as the intensity. In PAN-A2 the deformed shape suggests a change both of the direction of the compressed struts and in the

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FIGURE 16.16 Masonry deep beam. Left: degraded FRP
masonry interfaces patch for normal action (from 0—no degradation—to 1—full degradation) obtained through the nonlinear homogenized FE code. Right: shear damage at FRP
masonry interfaces (delamination, from 0—no degradation—to 1—full degradation).

failure mechanism. The deformed shape at collapse shows compression near the supports, shear under the load, and delamination of the diagonal strengthening. This is partially confirmed by the color map of damaged zones in the masonry interfaces reported again in Figs. 16.15 and 16.16 for shear and normal springs, respectively, and the delamination patch of the reinforcing strip—referred to as tangential FRP/masonry interface stresses—registered at the peak and depicted in Fig. 16.16. Assessment of the aforementioned nonlinear homogenization model in the case of curved structures is provided by analyzing a circular arch with buttresses and longitudinal strengthening, loaded with a horizontal action simulating an earthquake and already analyzed numerically by Mahini et al. (2007) in the presence and absence of strengthening. Analogous to the previous case, where available, mechanical properties experimentally determined for the constituent materials are adopted. In particular, in Mahini et al. (2007), a wide experimental characterization in compression on bricks and gypsum-clay prisms extracted from the original units as a part of the vaults is at disposal. In absence of specific experimentation on some parameters, literature indications are followed. The homogenization model utilized at the mesoscale was previously mentioned and again the interested reader is referred to Milani and Lourenc¸o (2013a,b) for a

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FIGURE 16.17 Semicircular arch with buttresses. Geometry, loading condition, and FE discretization adopted for the nonlinear numerical analyses.

comprehensive analysis of the parameters adopted in the structural simulations reviewed here. The aim of Mahini et al. (2007) was to have insight into the behavior of a typical existing roof vault diffused in a heritage complex building in Iran. The system of vaults was built in 1935 by adobe and clay bricks with clay mortar and gypsum
clay mortar, respectively. The vault has a circular shape with radius equal to 3.50 m (see Fig. 16.17) with a span L equal to 6.47 m. Buttresses have a height equal to 3.17 m. Piers and vault thicknesses are equal to 0.9 and 0.2 m, respectively. All geometrical dimensions together with the structural components of the arch are schematically shown in Fig. 16.17.

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While resistance to vertical gravity loads is typically reasonably good in this type of construction, the lateral performance is very poor and thus such arches need comprehensive seismic upgrading. For this reason, the strengthening intervention shown in Fig. 16.17 is numerically evaluated by Mahini et al. (2007), who modeled the structure with a smeared crack material, available within the standard commercial version of the FE code ANSYS (Stolarski et al., 2007). Piers are made of a different material with respect to the vault, being built with adobe and clay mortar. In the model, homogenization is obviously bypassed for the piers and an isotropic elasto-plastic material is utilized. Again, some experimental data (full stress
strain diagrams) in compression on new adobe piers may be collected from Mahini et al. (2007). Numerical simulations are performed by applying self-weight and an increasing lateral load constantly distributed along the height of piers, simulating roughly a seismic load proportional to the mass as shown in Fig. 16.17 and in agreement with Mahini et al. (2007). In order to investigate the seismic upgrading of the circular arch obtained through FRP strengthening, the structure is retrofitted with one strip of composite material placed at the extrados of the arch and two short strips on the surfaces of the piers subjected to tension, as shown in Fig. 16.17. The width of strip is equal to 20 cm. Unidirectional CFRP strips are used. The lateral behavior of the vault in terms of base shear-maximum horizontal displacement is depicted—in the presence and absence of strengthening—in Fig. 16.18A. In the absence of available experimental data, only a comparison with numerical results obtained by Mahini et al. (2007) and limit analysis collapse loads provided by a homogenized upper bound FE approach, (e.g., that reviewed previously in Chiozzi et al., 2018a, b) or similar without NURBS (e.g., Milani et al., 2008; Milani, 2015a) is possible here. In any case, again the global behavior seems in satisfactory agreement with alternative numerical procedures. Here, it is worth noting that the total lateral load-carrying capacity of the nonstrengthened vault is around 31.2 kN, whereas for the strengthened case it increases to a value equal to 40 kN. The retrofitting scheme proposed provides therefore a 30% increase in the load-carrying capacity, whereas the maximum horizontal displacement decreases (percentage difference of around 20%). This is completely in agreement with the FRP architecture, which aims for an increase in strength rather than ductility capacity. In Fig. 16.18B the deformed shape at peak obtained with the nonlinear homogenization approach reviewed here are represented for both the strengthened and the nonstrengthened case. Without FRP, the arch fails for the formation of relatively well-defined cylindrical hinges (H1, H2, H3, and H4), three located along the arch (H2, H3, and H4), and the latter (H1) at the base of the right pier. Hinge H3 is located near the center of the arch. This is not surprising because the vertical load is relatively small, and the structure

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FIGURE 16.18 Circular arch with buttresses. (A) Global load
displacement curves obtained numerically in presence and absence of reinforcement; (B) deformed shapes at peak provided.

with only the horizontal load applied is antisymmetric and the central section therefore exhibits a null precompression. The small axial force is due only to gravity loads and the section fails with very little bending (again due to vertical loads). The remaining two hinges on the arch are, as expected, in antisymmetric disposition. No prediction may be attempted for the piers because their precompression is sensibly higher. The position of the hinges is also well represented by the damage map on masonry interfaces depicted in Fig. 16.19. The strengthening is obviously placed in tensile zone on piers and at the extrados on the arch, in such a way to preclude the hinges opening. H3 remains near the middle span at the intrados because reinforcement is located in the compressed fiber of the section. The opening of this hinge seems more defined when compared to the nonstrengthened case. Again, H2 forms at the top edge on the right of the arch, but the damaged region diffuses considerably, as a consequence of the action of the strip that tends to preclude the extrados opening (see damage maps in Fig. 16.19). H4 moves from the arch

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FIGURE 16.19 Circular arch with buttresses. Top: masonry degraded interfaces patch for normal stress action (from 0—no degradation—to 1—full degradation). Bottom: FRP delamination patch for shear stress.

to the base of the left pier, because its opening is precluded by the presence of the FRP strip. This is a clear positive effect of strengthening with FRP strips, which tends also to diffuse damage near the base of both piers in correspondence of hinges H1 and H4, accompanied by a considerable delamination of the strips (Fig. 16.19). As can be noted again from Fig. 16.19 in the damage patch for FRP, the strip delaminates near the supports, and for the H2 hinge on the arch, in the tensile zone.

16.6 Conclusions This chapter reviewed two advanced numerical approaches for structural analysis in the nonlinear static range of FRP-reinforced masonries, namely a novel adaptive upper bound limit analysis and a nonlinear procedure with rigid elements based on sequential quadratic programming. Both tools have proven to be effective for the prediction of collapse (limit analysis) and beyond elasticity (nonlinear approach) of FRP-strengthened masonry. They

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require a preliminary homogenization of the masonry material at the mesoscale using a suitable homogenization approach. For dealing with the UB limit analysis procedure, a GA-NURBS-based general framework suitable for a mesh adaptation applied to curved masonry structures was discussed. As a matter of fact, any FRP-reinforced masonry vault can be geometrically represented by a NURBS parametric surface, and a NURBS mesh of the given surface can be generated. Each element of the mesh is a NURBS surface itself and can be idealized as a rigid body. An upper bound limit analysis formulation, taking into account the main characteristics of masonry material and FRP reinforcement, was reviewed, where internal dissipation was allowed exclusively along element interfaces. GA helps in progressively adjusting the shape of the NURBS elements to closely approximate the real failure mechanism that activates, thus providing good estimates of the load-carrying capacity. When dealing with the nonlinear approach, a simple model relying on FE discretization without mesh adaptation constituted by both rigid wedge elements (masonry) and rigid triangular elements (FRP) interconnected by nonlinear homogenized interfaces was reviewed. In such a framework, the step-by-step nonlinear problem can be solved using a sequential quadratic programming scheme, minimizing the constrained quadratic energy function. Both approaches have proven to be capable of well predicting the load-bearing capacity of any FRP-reinforced masonry structure (in particular, vaults of arbitrary shape, which are the most complex), with the nonlinear model having the additional features of accurately predicting initial stiffness, postpeak behavior, and displacements at failure. Both approaches have been benchmarked here through a number of numerical simulations applied to real FRP-reinforced masonry structures, for which experiments and/or alternative results obtained with other numerical approaches are available.

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