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strengthened masonry structures
Composites: Part B 36 (2005) 619–626 www.elsevier.com/locate/compositesb
Load carrying capacity of 2D FRP/strengthened masonry structures Luigi Ascione, Luciano Feo*, Fernando Fraternali Department of Civil Engineering, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy Received 29 November 2004; accepted 27 December 2004 Available online 14 July 2005
Abstract An adaptive discontinuous finite element model is formulated for limit analysis of masonry structures strengthened by fiber composites. The model is able to predict the effects of fracture damage and delamination on the load carrying capacity of the reinforced structures. A numerical investigation on the collapse mechanisms of masonry structures under plain strain/stress is presented, accounting for different mechanical properties of FRP–masonry interface and different placements of the reinforcement in the masonry structures. q 2005 Elsevier Ltd. All rights reserved. Keywords: B. Delamination; Masonry
1. Introduction Fiber reinforced plastic (FRP) materials are being considered for use in strengthening and repairing concrete and masonry structures due to their ease of application, noninvasiveness, lightweight, high strength, high stiffness, and excellent corrosion resistance. Economical and mechanical factors lead to the application of these materials in narrow strips, rather than full-width sheets. Several experimental and analytical studies have shown that FRP-retrofits can produce remarkable increases in load carrying capacity and out of plane strength in existing masonry structures [1–8]. Such a restoration technique must be used prudently, since faulty design of the reinforcing system may lead to premature failure modes and overstiffness of the structure. In particular, application of high strength uniaxial tensile reinforcements to a brittle substrate calls for considerable caution [8–11]. This paper presents a numerical study about the load carrying capacity of masonry structures reinforced by FRP materials. A collapse analysis of the reinforced structures is carried out using the kinematical method of limit analysis [5,6,12–14], accounting for associated inelastic behavior with respect to a given failure surface; small (or zero) tensile * Corresponding author. Tel.: C39 089 964044/46.
1359-8368/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2004.12.004
strength and finite compressive strength in masonry; FRP– masonry sliding. The external loads are split into a fixed quote (permanent loading) and a variable part (accidental loading), with the second one growing proportionally to a scalar parameter (load multiplier). The search for the collapse load, i.e. for the amplitude of the accidental loading which produces a state of collapse in the structure, is performed through a discontinuous adaptive finite element model, employing a two-level minimization procedure. Some benchmark problems are studied, referring to limit behavior of FRP-strengthened masonry vaults and walls.
2. Collapse of a strengthened masonry structure Let us consider a plane region U corresponding to the cross-section of a masonry structure in plane stress/strain conditions such as a wall, an arched structure and a structure covered with a barrel vault. We assume that such a structure is loaded by permanent forces with density p (per unit volume) and variable forces with density lq, l being a scalar multiplier. We also suppose that it has been strengthened with FRP plates or strips along a portion of its boundary, whose trace in the plane of U is a subset vUs of vU. Finally, we admit that masonry exhibits an associated inelastic behavior with respect to a fixed failure surface and that the FRP–masonry interface is characterized by
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2
-k2
_ remain unknown in Eq. (1), thus We let K (jump set of u) bringing the kinematical theorem back to a Free Discontinuity Problem [17]. Opening fractures in masonry are assumed to not dissipate energy [9,14,15], while sliding (Mode II) cracks on vUs can dissipate plastic energy. The FRP reinforcement impede the opening of fractures over the reinforced portion of the boundary. Thus, it influences the collapse behavior of the structure and increases its load carrying capacity.
fck
k1
-1
k1
1
fck
-1
3. Finite element model
-k2
Fig. 1. Masonry failure surface.
a perfectly plastic t–s law, t being the tangential bond stress, and s the interfacial slip. The failure surface adopted for masonry is shown in Fig. 1, where s1 and s2 denote the principal stresses, fck the uniaxial compressive strength, k1 the tensile vs compressive (uniaxial) strength ratio; and k2 the biaxial vs uniaxial compressive strength ratio. For the sake of simplicity, we disregard anisotropy induced in masonry by mortar joints and the softening behavior of the FRP–structure interface, which has been experimentally observed by several authors [1–3,8–9]. The assumption of masonry isotropy is often accepted in several practical applications [12–16], mainly in consideration of the disordered arrangement of brick elements, which is typically encountered in old masonry structures. We admit that inelastic strain L in masonry observes a normality rule with respect to the failure surface. In the above settings, the collapse multiplier lc of the variable loads can be computed through the kinematical theorem of limit analysis [5,6,12–14], which can be stated as follows 0 B lc Z min @K
ð
ð p$u_ dx C
_ u;K
UnK
UnK
_ uÞÞ _ dx C DðLð
ð
Let us introduce an adaptive triangulation of U (Fig. 2) and a piecewise linear approximation of u_ on Th, which is allowed to be discontinuous along mesh interfaces, under _ _ _ is the the condition that it results ½u$nZ d$n. Here, ½u _ in correspondence with the generic interface jump of ½u G, n the unit normal to G and d_ a non-negative scalar (detachment or crack opening). The triangulation Th is obtained by refining a first (primal) mesh (represented by solid lines in Fig. 2), in such a way that each triangle is split into four sub-elements. New edges are introduced (dashed edges in Fig. 2), and their extreme points are allowed to move along the edges of the primal mesh. The result is an adaptive triangulation [18]. Let us express the coordinates of movable nodes in terms of scalar variables x2[K1,1] and collect the nodal values of u_ into a vector v 2Rn (observe that nodes of adjacent triangles sharing same positions can exhibit different velocities). By introducing a suitable linearization (from outside) of the masonry failure surface in the 3D space of Cartesian stress components sx, sy, txy (Fig. 3), and fixing node placements, a finite element approximation of Eq. (1) leads
1 C _ dxA sðuÞ t_
vUS
(1) where: u_ 2C 0 ðU=KÞ is the velocity field corresponding to an admissible collapse mechanism of the structure, allowing for opening (Mode I) fractures over a one-dimensional compact subset K of U and interfacial FRP–masonry slip. _ uÞÞ _ is the rate of energy dissipation per unit volume DðLð in masonry. t is the bond strength of the FRP–masonry interface.
Fig. 2. Adaptive discontinuous finite element model.
L. Ascione et al. / Composites: Part B 36 (2005) 619–626
621
0,8m
p
6m
p
6m
4m
10m
6m
2m
4m
2m
13m
C
6m
10m
6m
Fig. 5. Cross-section of the structure.
Fig. 3. Linearization of masonry failure surface.
us to a Linear Programming Problem of the form lc0 ðxÞ Z min c$v v
subject to : bl %
Av % bu v
(2)
where A is a suitable m!n (sparse) matrix; c 2Rn ; bl ; bu 2 Rm (mZ number of linear constraints). Mesh adapting can then be introduced by dealing with a second-level minimization problem: lc Z min lc0 ðxÞ: x2½K1;1
(3)
In consideration that we are looking for the global minimum of kinematically admissible collapse multipliers, it is useful to employ an Evolution Strategy [19–21] to solve problem (3), which is strongly non-convex. Indeed, as
Fig. 4. Monumental masonry structure covered with a barrel vault.
opposed to gradient methods, such strategies are able to handle optimization problems with several local optima. It is worthwhile noticing that crack and sliding patterns at collapse become part of the solution of the proposed procedure. The adaptive discontinuous finite element model allows one to perform optimization with respect to both the collapse mechanism and the collapse multiplier of the structure, as opposed to fixed discontinuous element models, where the crack pattern at collapse is forced to adapt to the ‘skeleton’ of a given mesh. In all the examples given in Section 4, the adopted evolutionary algorithms [19,20] reached to a stable solution within a reasonable number of generations (about ten generations were needed to obtain a stable value of the collapse multiplier).
4. Numerical results Hereafter, we present some applications of our adaptive discontinuous finite element model to the computation of
Fig. 6. Strengthening by FRP strips.
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Table 1 Material properties of Example 1 Material/property Masonry Unit weight (kN/m3) Compressive strength (MPa) k1 k2 FRP Tensile strength (GPa) Thickness (mm) Width (m) Interface Bond strength (MPa)
Value 17 2.2 0 1.25 4.89 0.177 0.40 0.01O0.1
the ultimate load and the collapse mechanism of two-dimensional FRP/reinforced masonry structures. The given results have qualitative nature and aim to highlight the applicative potential of the proposed model. Experimental verification is left to future works, also in consideration of the fact that FRP retrofit of masonry structures is a relatively recent reinforcing technique and that up to now limited experimental results are available in the literature. Adopted values of the FRP/masonry bond strength correspond to multiples of a characteristic value of the shear strength of real masonry structures (0.1 MPa). Very low values of such strength have been introduced to reproduce situations in which the interface has not been adequately prepared before the application of FRP elements. 4.1. FRP/strengthening of a barrel vault Let us firstly deal with the collapse analysis of the monumental masonry structure shown in Figs. 4–6, consisting of a barrel vault resting on thick buttresses [20]. The loading condition is represented by the self-weight p of the structure and horizontal forces lp. In this example, indeed, the base value of variable forces has been identified
Fig. 7. Collapse mechanism in absence of FRP strengthening.
Fig. 8. Collapse mechanisms of the externally reinforced structure, for two different values of the bond strength t.
with the value of permanent forces, since the first are thought of as static horizontal actions corresponding to a seismic loading. The vault is strengthened with FRP strips bonded either at the extrados, as shown in Fig. 6, or at the intrados. Each strip is 10 cm wide and has a 25 cm pitch (cf. real examples given in [2–6]), giving a total of 40 cm of fiber reinforcement for each meter of the longitudinal development of the structure. Here and in the next example, it is assumed that the foundation of the structure remains rigid and undamaged until collapse. Material properties used in numerical calculations are listed in Table 1. Carbon fibers embedded in epoxy resin (CFRP) [6] were considered. Masonry failure surface was approximated with a 36-faces polyhedron as shown in Fig. 3. We employed the discontinuous self-adapting finite element model presented in Section 3, in order to obtain
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F3
F2
F1
Fig. 9. Collapse mechanism of the internally reinforced structure 0:1 MPaÞ. ðtZ
estimates of the collapse multiplier lc of the horizontal forces. A 1.0 m long slice of the structure was analyzed, for a total FRP-width of 40 cm. The collapse mechanism obtained for the unreinforced structure in shown in Fig. 7 (lcZ0.1053). It corresponds to the development of four opening hinges (cracks) in the masonry, one of which is located in a buttress and the other three in the vault (‘semi-global’ mechanism). Crushing of masonry near the crack tips was observed. The result in Fig. 7 corresponds well with that obtained in [22] for the same problem (lcZ0.1056 in [22] for infinite compressive strength of masonry). Mesh adapting during the optimization procedure lead the finite element model to predict stress concentration in masonry around crack tips. As a second case, we considered a FRP reinforcement at the extrados of the vault (cf. Fig. 6) and modeled the strengthened structure for two different values of the bond strength t. Figs. 8a,b shows the collapse mechanisms corresponding 0:1 and tZ 0:01 MPa, respectively. to tZ 0:1 MPaÞ, We obtained lcZ0.1644 in the first case ðtZ which corresponds to an increase of 56% in the ultimate load carrying capacity, with respect to the unstrengthened structure. It can be observed (Fig. 8a) that the reinforced structure exhibits a ‘frame’ collapse mechanism in such a case (two opening hinges in the vault and two hinges in the buttresses). Fracturing of masonry near the right buttress was observed. 0:01 MPa shows considerable differThe case of tZ ences, since such a small value of bond strength leads to premature FRP–masonry sliding near vault-buttress connections (Fig. 8b). Now, three opening hinges appear in the vault at collapse, and the increase in load carrying capacity reaches 43% (lcZ0.1504). The last situation we examined deals with a placement of the FRP reinforcement on the inner side of the structure as
Fig. 10. Masonry wall with openings subjected to fixed vertical loads and variable horizontal forces.
shown in Fig. 9. Such a strengthening technique proved to be less effective than the previous one, being associated with masonry ripping near the middle span of the vault at collapse. It produced a reduced increase in load carrying
Fig. 11. Mesh adopted for Example 2.
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capacity (only 5%), leading to lcZ0.1108. The collapse mechanism is again ‘semi-global’ and shows similar behavior to that exhibited by the unreinforced structure (Fig. 7). The external reinforcement performs better than the inner one due to the fact that the collapse mechanism of the unreinforced structure shows two cracks at the extrados of the vault and only one at the intrados (Fig. 7). 4.2. FRP/plating of the openings of a masonry wall The second example we examined concerns a three storey masonry wall subjected to fixed vertical loads and variable horizontal forces (see Fig. 10). The wall is made up of tufe stones having 1.0 m thickness (constant over the height) and 18 kN/m3 unit weight. A permanent loading of 7.5 kN/m is acting in correspondence with each storey level. The base values of horizontal forces are: F1Z33.82 kN; F2Z45.58 kN; F3Z 70.31 kN. Upon assuming rigid behavior of masonry and employing the mesh of Fig. 11 (with no adaptation), upper bound estimates of collapse multiplier (lc) were obtained solving LP problem (2). Horizontal forces were uniformly distributed over the edges placed at the storey levels. Fig. 12 shows the collapse mechanism obtained for the unreinforced wall (lcZ3.9827).
Fig. 12. Collapse mechanism of the unreinforced wall (lcZ3.9827).
Table 2 Material properties of Example 2 Material/property Masonry Unit weight (kN/m3) Rigid behavior FRP Tensile strength (GPa) Thickness (mm) Width (m) Interface Bond strength (MPa) Steel ties Plastic axial force (kN) Steel flitched beams Plastic bending moment
Value 18
4.89 0.177 0.50 0.001O0.1 350 450
As a first strengthening technique, we considered the placement of steel ties at the base of arched openings and steel flitched beams over rectangular openings (cf. Table 2). This kind of reinforcement lead to an increase of about 58% in the load carrying capacity of the wall (lcZ6.2945, see Fig. 13). Next, we dealt with a FRP-plating of all the openings, along three sides (top, right and left), considering two different values of the bond strength as shown in Figs 14 and 15. Fig. 14 illustrates the collapse mechanism corresponding to a sufficiently large value of the bond strength
Fig. 13. Collapse mechanism of the wall strengthened with steel ties and flitched beams (lcZ6.2945).
L. Ascione et al. / Composites: Part B 36 (2005) 619–626
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0:1 MPaÞ. It can be seen that a great increase of ðtZ the load carrying capacity is produced by the FRP retrofit in such a case lcZ9.2505, C133% with respect to the unreinforced wall and C47% with respect to the steelreinforced wall). 0:001 MPa, as Differently, for a very low bond strength tZ in the previous example, the efficiency of the FRP reinforcement is dramatically reduced lcZ4.1978, only C5.4% with respect to the unreinforced wall). Indeed, in such a case, diffused FRP debonding is observed at collapse (Fig. 15).
5. Concluding remarks
0:1 MPa (lcZ9.2505). Fig. 14. FRP-strengthened wall—tZ
The finite element model presented in this work can be used to predict the ultimate load carrying capacity of FRP reinforced masonry structures. It accounts for the main phenomena which characterize the limit behavior of such structures (crushing and fracturing of masonry, interface debonding and ripping, stabilizing effects played by masonry self-weight). The model has been employed to carry out a numerical study regarding the increase in the loading carrying capacity produced by FRP/strengthening. The given results highlight the importance of curing the properties of a FRP–masonry interface to prevent premature failure mechanisms (low bond strength). The use of an adaptive discontinuous finite element model allowed us to obtain detailed information about the collapse mechanism (location and arrangement of the crack pattern; material crushing and ripping) of the reinforced structure through direct minimization of the collapse multiplier. A 3D generalization of the proposed model, experimental verification and application to a wide spectrum of reinforcing techniques will be presented in future works.
Acknowledgements The supports of CNR Grant ‘Models for damage prediction of civil constructions and infrastructures’ (Italian National Research Council) and MIUR Grant ‘Computational methods for minimization of non-convex energies with interface terms: application to brittle and no-tension solids’ (Italian Ministry of University and Research) are gratefully acknowledged.
References
0:001 MPa (lcZ4.1978). Fig. 15. FRP-strengthened wall—tZ
[1] Saadatmanesh H. Extending service life of concrete and masonry structures with fiber composites. Constr Building Mat 1997;11: 327–35. [2] Corradi M, Borri A, Vignoli A. Strengthening techniques tested on masonry structures struck by the Umbria–Marche earthquake of 1997–1998. Constr Building Mat 2002;16:229–39.
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[3] Valluzzi MR, Tinazzi D, Modena C. Shear behaviour of masonry panels strengthened by FRP laminates, k. Constr Building Mat 2002; 16:409–16. [4] Briccoli Bati S, Rapallini M, Tralli A. Strengthening of masonry vaulted structures using F.R.P. overlays. Proceedings of ARCH’01, Paris 2001. [5] Baratta A, Corbi O. Programmazione lineare ed analisi limite di strutture ad arco fibrorinforzate. Proceedings of AIAS 2003, Salerno, Italy, CD-ROM 2003. [6] Baratta A, Corbi O. Valutazione dell’efficacia delle fibre di carbonio nella riparazione di archi murari: un confronto numerico/sperimentale. Proceedings of AIMETA’03, Ferrara, Italy, CD-ROM 2003. [7] Cecchi A, Milani G, Tralli A. A simple micro-mechanical model for the homogeneized constitutive equations of masonry. Proceedings of AIMETA’03, Ferrara, Italy, CD-ROM 2003. [8] Lofti H, Shing P. Interface model applied fracture of masonry structures. ASCE J Struct Eng 1994;102:63–80. [9] Savoia M, Ferracuti B, Mazzotti C. Una legge di interfaccia non lineare per placcaggi con lamine in FRP. Proceedings of AIMETA’03, Ferrara, Italy, CD-ROM 2003. [10] Ascione L, Feo L. Modelling of composites/concrete interface of RC beams strengthened with composite laminates. Compos Part B: Eng 2000;31:535–40. [11] Ascione L, Berardi VP, Feo L, Mancusi G. A numerical evaluation of the interlaminar stress state in externally FRP plated RC beams. Compos Part B: Eng 2005;36:83–90. [12] Heyman J. The stone skeleton. Cambridge, UK: Cambridge University Press; 1995.
[13] Del Piero G. Limit analysis and no-tension materials. Int J Plasticity 1998;14:259–71. [14] Lucchesi M, Padovani C, Pasquinelli G, Zani N. On the collapse of masonry arches. Meccanica 1997;32:327–46. [15] Giacquinta M, Giusti G. Researches on the equilibrium of masonry structures. Arch Rat Mech Analysis 1985;88:359–92. [16] Del Piero G. Constitutive equations and compatibility of the external loads for linear elastic masonry-like materials. Meccanica 1989;24: 150–62. [17] De Giorgi E. Un nuovo funzionale del calcolo delle variazioni’. Atti Accad Naz Lincei (Rend Cl Sci Fis Mat Natur) 1988;82:199–210. [18] Negri M. A finite element approximation of the Griffith’s model in fracture mechanics’. Numer Math 2003;95:653–87. [19] De Falco I, Della Cioppa A, Iazzetta A, Tarantino E. Evolutionary algorithms for aerofoil design. Int J Comput Fluid Dyn 1998;11:51–77 (OPA (Overseas Publisher Association) N.V., Gordon and Beach Science Publisher). [20] De Falco I, Del Balio R, Della Cioppa A, Tarantino E. Investigation a parallel breeder generic algorithm on the inverse aerodynamic design. In: Voigt HM, Ebeling W, Rechemberg I, Schwefel HP, editors. Lecture note in computer science 1141—parallel problem solving from nature IV. Berlin: Springer; 1996. p. 982–91. [21] Lagaros ND, Papadrakis M, Kokassolakis G. Structural optimization using evolutionary algorithms. Comput Struct 2002;80:571–89. [22] Como M, Abbruzzese D, Lanni G. Some results on the strength evaluation of vaulted masonry structures in architectural studies. Materials and Analysis, STREMA 95. Boston: Computational Mechanics Publications; 1995.
Load carrying capacity of 2D FRP/strengthened masonry structures Luigi Ascione, Luciano Feo*, Fernando Fraternali Department of Civil Engineering, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy Received 29 November 2004; accepted 27 December 2004 Available online 14 July 2005
Abstract An adaptive discontinuous finite element model is formulated for limit analysis of masonry structures strengthened by fiber composites. The model is able to predict the effects of fracture damage and delamination on the load carrying capacity of the reinforced structures. A numerical investigation on the collapse mechanisms of masonry structures under plain strain/stress is presented, accounting for different mechanical properties of FRP–masonry interface and different placements of the reinforcement in the masonry structures. q 2005 Elsevier Ltd. All rights reserved. Keywords: B. Delamination; Masonry
1. Introduction Fiber reinforced plastic (FRP) materials are being considered for use in strengthening and repairing concrete and masonry structures due to their ease of application, noninvasiveness, lightweight, high strength, high stiffness, and excellent corrosion resistance. Economical and mechanical factors lead to the application of these materials in narrow strips, rather than full-width sheets. Several experimental and analytical studies have shown that FRP-retrofits can produce remarkable increases in load carrying capacity and out of plane strength in existing masonry structures [1–8]. Such a restoration technique must be used prudently, since faulty design of the reinforcing system may lead to premature failure modes and overstiffness of the structure. In particular, application of high strength uniaxial tensile reinforcements to a brittle substrate calls for considerable caution [8–11]. This paper presents a numerical study about the load carrying capacity of masonry structures reinforced by FRP materials. A collapse analysis of the reinforced structures is carried out using the kinematical method of limit analysis [5,6,12–14], accounting for associated inelastic behavior with respect to a given failure surface; small (or zero) tensile * Corresponding author. Tel.: C39 089 964044/46.
1359-8368/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2004.12.004
strength and finite compressive strength in masonry; FRP– masonry sliding. The external loads are split into a fixed quote (permanent loading) and a variable part (accidental loading), with the second one growing proportionally to a scalar parameter (load multiplier). The search for the collapse load, i.e. for the amplitude of the accidental loading which produces a state of collapse in the structure, is performed through a discontinuous adaptive finite element model, employing a two-level minimization procedure. Some benchmark problems are studied, referring to limit behavior of FRP-strengthened masonry vaults and walls.
2. Collapse of a strengthened masonry structure Let us consider a plane region U corresponding to the cross-section of a masonry structure in plane stress/strain conditions such as a wall, an arched structure and a structure covered with a barrel vault. We assume that such a structure is loaded by permanent forces with density p (per unit volume) and variable forces with density lq, l being a scalar multiplier. We also suppose that it has been strengthened with FRP plates or strips along a portion of its boundary, whose trace in the plane of U is a subset vUs of vU. Finally, we admit that masonry exhibits an associated inelastic behavior with respect to a fixed failure surface and that the FRP–masonry interface is characterized by
620
L. Ascione et al. / Composites: Part B 36 (2005) 619–626
2
-k2
_ remain unknown in Eq. (1), thus We let K (jump set of u) bringing the kinematical theorem back to a Free Discontinuity Problem [17]. Opening fractures in masonry are assumed to not dissipate energy [9,14,15], while sliding (Mode II) cracks on vUs can dissipate plastic energy. The FRP reinforcement impede the opening of fractures over the reinforced portion of the boundary. Thus, it influences the collapse behavior of the structure and increases its load carrying capacity.
fck
k1
-1
k1
1
fck
-1
3. Finite element model
-k2
Fig. 1. Masonry failure surface.
a perfectly plastic t–s law, t being the tangential bond stress, and s the interfacial slip. The failure surface adopted for masonry is shown in Fig. 1, where s1 and s2 denote the principal stresses, fck the uniaxial compressive strength, k1 the tensile vs compressive (uniaxial) strength ratio; and k2 the biaxial vs uniaxial compressive strength ratio. For the sake of simplicity, we disregard anisotropy induced in masonry by mortar joints and the softening behavior of the FRP–structure interface, which has been experimentally observed by several authors [1–3,8–9]. The assumption of masonry isotropy is often accepted in several practical applications [12–16], mainly in consideration of the disordered arrangement of brick elements, which is typically encountered in old masonry structures. We admit that inelastic strain L in masonry observes a normality rule with respect to the failure surface. In the above settings, the collapse multiplier lc of the variable loads can be computed through the kinematical theorem of limit analysis [5,6,12–14], which can be stated as follows 0 B lc Z min @K
ð
ð p$u_ dx C
_ u;K
UnK
UnK
_ uÞÞ _ dx C DðLð
ð
Let us introduce an adaptive triangulation of U (Fig. 2) and a piecewise linear approximation of u_ on Th, which is allowed to be discontinuous along mesh interfaces, under _ _ _ is the the condition that it results ½u$nZ d$n. Here, ½u _ in correspondence with the generic interface jump of ½u G, n the unit normal to G and d_ a non-negative scalar (detachment or crack opening). The triangulation Th is obtained by refining a first (primal) mesh (represented by solid lines in Fig. 2), in such a way that each triangle is split into four sub-elements. New edges are introduced (dashed edges in Fig. 2), and their extreme points are allowed to move along the edges of the primal mesh. The result is an adaptive triangulation [18]. Let us express the coordinates of movable nodes in terms of scalar variables x2[K1,1] and collect the nodal values of u_ into a vector v 2Rn (observe that nodes of adjacent triangles sharing same positions can exhibit different velocities). By introducing a suitable linearization (from outside) of the masonry failure surface in the 3D space of Cartesian stress components sx, sy, txy (Fig. 3), and fixing node placements, a finite element approximation of Eq. (1) leads
1 C _ dxA sðuÞ t_
vUS
(1) where: u_ 2C 0 ðU=KÞ is the velocity field corresponding to an admissible collapse mechanism of the structure, allowing for opening (Mode I) fractures over a one-dimensional compact subset K of U and interfacial FRP–masonry slip. _ uÞÞ _ is the rate of energy dissipation per unit volume DðLð in masonry. t is the bond strength of the FRP–masonry interface.
Fig. 2. Adaptive discontinuous finite element model.
L. Ascione et al. / Composites: Part B 36 (2005) 619–626
621
0,8m
p
6m
p
6m
4m
10m
6m
2m
4m
2m
13m
C
6m
10m
6m
Fig. 5. Cross-section of the structure.
Fig. 3. Linearization of masonry failure surface.
us to a Linear Programming Problem of the form lc0 ðxÞ Z min c$v v
subject to : bl %
Av % bu v
(2)
where A is a suitable m!n (sparse) matrix; c 2Rn ; bl ; bu 2 Rm (mZ number of linear constraints). Mesh adapting can then be introduced by dealing with a second-level minimization problem: lc Z min lc0 ðxÞ: x2½K1;1
(3)
In consideration that we are looking for the global minimum of kinematically admissible collapse multipliers, it is useful to employ an Evolution Strategy [19–21] to solve problem (3), which is strongly non-convex. Indeed, as
Fig. 4. Monumental masonry structure covered with a barrel vault.
opposed to gradient methods, such strategies are able to handle optimization problems with several local optima. It is worthwhile noticing that crack and sliding patterns at collapse become part of the solution of the proposed procedure. The adaptive discontinuous finite element model allows one to perform optimization with respect to both the collapse mechanism and the collapse multiplier of the structure, as opposed to fixed discontinuous element models, where the crack pattern at collapse is forced to adapt to the ‘skeleton’ of a given mesh. In all the examples given in Section 4, the adopted evolutionary algorithms [19,20] reached to a stable solution within a reasonable number of generations (about ten generations were needed to obtain a stable value of the collapse multiplier).
4. Numerical results Hereafter, we present some applications of our adaptive discontinuous finite element model to the computation of
Fig. 6. Strengthening by FRP strips.
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L. Ascione et al. / Composites: Part B 36 (2005) 619–626
Table 1 Material properties of Example 1 Material/property Masonry Unit weight (kN/m3) Compressive strength (MPa) k1 k2 FRP Tensile strength (GPa) Thickness (mm) Width (m) Interface Bond strength (MPa)
Value 17 2.2 0 1.25 4.89 0.177 0.40 0.01O0.1
the ultimate load and the collapse mechanism of two-dimensional FRP/reinforced masonry structures. The given results have qualitative nature and aim to highlight the applicative potential of the proposed model. Experimental verification is left to future works, also in consideration of the fact that FRP retrofit of masonry structures is a relatively recent reinforcing technique and that up to now limited experimental results are available in the literature. Adopted values of the FRP/masonry bond strength correspond to multiples of a characteristic value of the shear strength of real masonry structures (0.1 MPa). Very low values of such strength have been introduced to reproduce situations in which the interface has not been adequately prepared before the application of FRP elements. 4.1. FRP/strengthening of a barrel vault Let us firstly deal with the collapse analysis of the monumental masonry structure shown in Figs. 4–6, consisting of a barrel vault resting on thick buttresses [20]. The loading condition is represented by the self-weight p of the structure and horizontal forces lp. In this example, indeed, the base value of variable forces has been identified
Fig. 7. Collapse mechanism in absence of FRP strengthening.
Fig. 8. Collapse mechanisms of the externally reinforced structure, for two different values of the bond strength t.
with the value of permanent forces, since the first are thought of as static horizontal actions corresponding to a seismic loading. The vault is strengthened with FRP strips bonded either at the extrados, as shown in Fig. 6, or at the intrados. Each strip is 10 cm wide and has a 25 cm pitch (cf. real examples given in [2–6]), giving a total of 40 cm of fiber reinforcement for each meter of the longitudinal development of the structure. Here and in the next example, it is assumed that the foundation of the structure remains rigid and undamaged until collapse. Material properties used in numerical calculations are listed in Table 1. Carbon fibers embedded in epoxy resin (CFRP) [6] were considered. Masonry failure surface was approximated with a 36-faces polyhedron as shown in Fig. 3. We employed the discontinuous self-adapting finite element model presented in Section 3, in order to obtain
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F3
F2
F1
Fig. 9. Collapse mechanism of the internally reinforced structure 0:1 MPaÞ. ðtZ
estimates of the collapse multiplier lc of the horizontal forces. A 1.0 m long slice of the structure was analyzed, for a total FRP-width of 40 cm. The collapse mechanism obtained for the unreinforced structure in shown in Fig. 7 (lcZ0.1053). It corresponds to the development of four opening hinges (cracks) in the masonry, one of which is located in a buttress and the other three in the vault (‘semi-global’ mechanism). Crushing of masonry near the crack tips was observed. The result in Fig. 7 corresponds well with that obtained in [22] for the same problem (lcZ0.1056 in [22] for infinite compressive strength of masonry). Mesh adapting during the optimization procedure lead the finite element model to predict stress concentration in masonry around crack tips. As a second case, we considered a FRP reinforcement at the extrados of the vault (cf. Fig. 6) and modeled the strengthened structure for two different values of the bond strength t. Figs. 8a,b shows the collapse mechanisms corresponding 0:1 and tZ 0:01 MPa, respectively. to tZ 0:1 MPaÞ, We obtained lcZ0.1644 in the first case ðtZ which corresponds to an increase of 56% in the ultimate load carrying capacity, with respect to the unstrengthened structure. It can be observed (Fig. 8a) that the reinforced structure exhibits a ‘frame’ collapse mechanism in such a case (two opening hinges in the vault and two hinges in the buttresses). Fracturing of masonry near the right buttress was observed. 0:01 MPa shows considerable differThe case of tZ ences, since such a small value of bond strength leads to premature FRP–masonry sliding near vault-buttress connections (Fig. 8b). Now, three opening hinges appear in the vault at collapse, and the increase in load carrying capacity reaches 43% (lcZ0.1504). The last situation we examined deals with a placement of the FRP reinforcement on the inner side of the structure as
Fig. 10. Masonry wall with openings subjected to fixed vertical loads and variable horizontal forces.
shown in Fig. 9. Such a strengthening technique proved to be less effective than the previous one, being associated with masonry ripping near the middle span of the vault at collapse. It produced a reduced increase in load carrying
Fig. 11. Mesh adopted for Example 2.
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capacity (only 5%), leading to lcZ0.1108. The collapse mechanism is again ‘semi-global’ and shows similar behavior to that exhibited by the unreinforced structure (Fig. 7). The external reinforcement performs better than the inner one due to the fact that the collapse mechanism of the unreinforced structure shows two cracks at the extrados of the vault and only one at the intrados (Fig. 7). 4.2. FRP/plating of the openings of a masonry wall The second example we examined concerns a three storey masonry wall subjected to fixed vertical loads and variable horizontal forces (see Fig. 10). The wall is made up of tufe stones having 1.0 m thickness (constant over the height) and 18 kN/m3 unit weight. A permanent loading of 7.5 kN/m is acting in correspondence with each storey level. The base values of horizontal forces are: F1Z33.82 kN; F2Z45.58 kN; F3Z 70.31 kN. Upon assuming rigid behavior of masonry and employing the mesh of Fig. 11 (with no adaptation), upper bound estimates of collapse multiplier (lc) were obtained solving LP problem (2). Horizontal forces were uniformly distributed over the edges placed at the storey levels. Fig. 12 shows the collapse mechanism obtained for the unreinforced wall (lcZ3.9827).
Fig. 12. Collapse mechanism of the unreinforced wall (lcZ3.9827).
Table 2 Material properties of Example 2 Material/property Masonry Unit weight (kN/m3) Rigid behavior FRP Tensile strength (GPa) Thickness (mm) Width (m) Interface Bond strength (MPa) Steel ties Plastic axial force (kN) Steel flitched beams Plastic bending moment
Value 18
4.89 0.177 0.50 0.001O0.1 350 450
As a first strengthening technique, we considered the placement of steel ties at the base of arched openings and steel flitched beams over rectangular openings (cf. Table 2). This kind of reinforcement lead to an increase of about 58% in the load carrying capacity of the wall (lcZ6.2945, see Fig. 13). Next, we dealt with a FRP-plating of all the openings, along three sides (top, right and left), considering two different values of the bond strength as shown in Figs 14 and 15. Fig. 14 illustrates the collapse mechanism corresponding to a sufficiently large value of the bond strength
Fig. 13. Collapse mechanism of the wall strengthened with steel ties and flitched beams (lcZ6.2945).
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0:1 MPaÞ. It can be seen that a great increase of ðtZ the load carrying capacity is produced by the FRP retrofit in such a case lcZ9.2505, C133% with respect to the unreinforced wall and C47% with respect to the steelreinforced wall). 0:001 MPa, as Differently, for a very low bond strength tZ in the previous example, the efficiency of the FRP reinforcement is dramatically reduced lcZ4.1978, only C5.4% with respect to the unreinforced wall). Indeed, in such a case, diffused FRP debonding is observed at collapse (Fig. 15).
5. Concluding remarks
0:1 MPa (lcZ9.2505). Fig. 14. FRP-strengthened wall—tZ
The finite element model presented in this work can be used to predict the ultimate load carrying capacity of FRP reinforced masonry structures. It accounts for the main phenomena which characterize the limit behavior of such structures (crushing and fracturing of masonry, interface debonding and ripping, stabilizing effects played by masonry self-weight). The model has been employed to carry out a numerical study regarding the increase in the loading carrying capacity produced by FRP/strengthening. The given results highlight the importance of curing the properties of a FRP–masonry interface to prevent premature failure mechanisms (low bond strength). The use of an adaptive discontinuous finite element model allowed us to obtain detailed information about the collapse mechanism (location and arrangement of the crack pattern; material crushing and ripping) of the reinforced structure through direct minimization of the collapse multiplier. A 3D generalization of the proposed model, experimental verification and application to a wide spectrum of reinforcing techniques will be presented in future works.
Acknowledgements The supports of CNR Grant ‘Models for damage prediction of civil constructions and infrastructures’ (Italian National Research Council) and MIUR Grant ‘Computational methods for minimization of non-convex energies with interface terms: application to brittle and no-tension solids’ (Italian Ministry of University and Research) are gratefully acknowledged.
References
0:001 MPa (lcZ4.1978). Fig. 15. FRP-strengthened wall—tZ
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