Experimental validation of a proposed numerical model ... - Group HMS
Historical Constructions, P.B. Lourenço, P. Roca (Eds.), Guimarães, 2001
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Experimental validation of a proposed numerical model for the FRP consolidation of masonry arches S. Briccoli Bati and L. Rovero University of Florence, Dipartimento di Costruzioni, Firenze, Italy
ABSTRACT: In planning the restoration and consolidation of historical buildings one must assess the effects on their structural behavior of applying advanced materials and technologies to elements made with traditional ones. The present paper deals with the consolidation of masonry arches through application of Carbon Fiber Reinforced Polymers (CFRP). A numerical model for arches reinforced with FRP sheets is proposed, and its validity checked through experimental trials. The numerical model represents the arch as a rigid assembly of blocks joined by elastic connecting rods with bounded tensile strength, which simulate the mortar joint, and tensionresistant connecting rods on the intrados and/or extrados, which simulate the FPR reinforcement. The numerical model has been checked experimentally through a series of tests on 1:2 scale models of brick arches subjected to the application of a load concentrated at the keystone.
1 INTRODUCTION Safeguarding the world's historical buildings requires defining appropriate methodologies for their restoration and consolidation. In defining such methodologies, one must account for, on the one hand, structural safety, including seismic hazard, and on the other, respect for the original architecture from both the aesthetic as well as structural perspectives. The consolidation techniques recommended and widely used in recent years for masonry structures include the addition of reinforced-concrete floors or centering to the extrados of vaulted structures. These, however, represent highly invasive and irreversible operations. In fact, apart from altering the original state and function of the structures under restoration, such methods often introduce extraneous elements that are incompatible with the pre-existing conventional materials and techniques, so much so that, at least in some cases, their application winds up contributing decisively to ruining the structure they were supposed to preserve. With regard to the many issues associated with historical building maintenance and restoration, the interest of workers in the sector has increasingly turned to the development of advanced technologies and the adoption of innovative materials. Many innovative materials are able to satisfy mechanical requirements not met by traditional materials and technologies. Of these, composite materials represent a particularly valuable group and have already been applied widely in aeronautics, mechanics and sporting equipment. The composite materials most often used in the fields of civil engineering and restoration are the FRP (Fiber Reinforced Polymer).
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FRP are composite material make by the union of fibers and polymeric matrix. The fibers are the strong components and are short and random arranged or long and in one or more direction arranged. The fibers are either glasses fibers (GFRP, Glasses Fiber Reinforced Polymer), either aramidic fibers (AFRP, Aramidic Fiber Reinforced Polymer), or carbon fibers (CFRP, Carbon Fiber Reinforced Polymer). The matrix protects the fibers, distributes the tensions and gives shape to the composite. The more used matrices are the thermosetting polymeric matrices. The principal characteristics of FRP are their high strength and stiffness relative to their weight, their considerable volumetric stability, good resistance to chemical action and low electrical conductivity. As they are anisotropic, they can be custom designed for specific tasks by arranging the fibers in such a way as to provide optimal load-bearing capacity for the particular loading conditions. The use of sheets of composite material in the form of cladding, applied to either the intrados or the extrados of vaulted structures, could obviate at least some of the inherent drawbacks of the consolidation techniques currently in use. In order to assess the efficiency of such reinforcement techniques, we conducted an experimental analysis on 1:2 scale models of brick arches consolidated by application of CFRP sheets and subjected to the application of a load concentrated at the keystone. The experimental results obtained were then utilized to check a numerical model that simulates the behavior of masonry arches reinforced with FRP cladding. The arch is modeled as a series of rigid blocks joined at their interfaces by elastic connecting rods. In particular the voussoir is modeled as rigid block; the mortar-joint is modeled as a series of elastic rods with bounded tensile strength; the FRP reinforcement, in correspondence to the intrados and extrados, is simulated by elastic tension-resistant connecting rods endowed with suitable extensional stiffness.
2. NUMERICAL MODEL
2.1 Description In Briccoli Bati et al. (1998) a numerical model is introduced for calculations on masonry arches, under the assumptions that the voussoirs are rigid and the interfaces elastic-cracking. Such modeling furnishes results in perfect agreement with experimental results and able to accurately interpret actual cases. In the present work the numerical model has been modified in order to enable its application to calculations relative to arches reinforced with FRP sheets placed on the intrados and/or extrados.
A
RIGID BLOCK
B
ELASTIC NO-TENSION ROD
C
ELASTIC ROD
D
ELASTIC ROD
Figure 1: Numerical model.
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The proposed model conceives of the reinforced arch as made up of rigid blocks fixed to the interface by three types of appropriately rigid elastic connecting rods. In Figure 1 the connecting rods indicated by the letters A and B simulate the mortar joints: more precisely, type A joints, tangent to the arch axis, have bounded tensile C and absorb the normal component, while type B, orthogonal to the arch axis, resists both tension and compression and absorbs the shear component. Connecting rods C, resistant alone to tension, are set on the intrados or extrados and represent the FRP reinforcement. The rods A with bonded tensile strength were eliminated, when the internal force is major or equal to ultimate tensile strength. The elimination of the rods fits the reduction of section because there is a fracture due to the no-tension strength of the masonry. The stiffness of the rod B corresponding to eliminated rods A is reduced in proportion to the number of eliminated rods A in the section. The mechanical parameters of the model are the elastic modulus of the rods A and B, the ultimate tensile strength of the rods A and the elastic modulus of the rods C. The elastic modulus and the tension limit of A rods depend on mechanical characteristics of joint mortar. The elastic modulus of C rods depend on mechanical characteristics of the FRP reinforcement. The elastic modulus of the rods B depends on the mechanical characteristics of joint mortar and on geometric characteristics of the section by means of following formula:
Nl B λTt = GAsez ErodB ArodB with λ shear factor, T tangential force, t joint thickness, G elastic tangential modulus of the mortar, As e z surface of the cross section of the structure, N normal force, lB length of rod B equal to height of structure section, ArodB surface of the cross section of the rod B. E rodB is determined imposing that the longitudinal deformation of the rod B with normal force is equal to the deformation produced by the tangential force T (equal to normal force N ) in the section of the structure. 2.2 The no-tension problem’s equations The constitutive model for a stucture consisted of rigid blocks and elastic roods with bounded tensile streght is derived from linear elastic model through the correction of the static solution and through the following determination of the kinematic solution consistent with the new static solution. The static solution is determined adding selfequilibrated stress to elastic stress to annul the not acceptable tensile tress. The kinematic solution is determined considering the selfequilibrated stress as internal forces resulted from applied strains that give validity to kinematic equations. The applied strains are as fractures caused of the bounded tensile strenght. In particular the equation of the problems are: for a structure made up of n rigid bodies connected by m elastic connecting rods (fig.2), of which z with bounded tensile strength lim TEN , the equations for the static and kinematic problem are AX + F = O A T x + KX + ∆ = O
(1)
X Z ≤ lim TEN
where: A ∈ R m×3 n is the equilibrium operator or topological matrix; X ∈R m is the vector of the normal forces in the connecting rods; z are the rods absorbing the normal component of the mortar joints and therefore with bounded tensile strength ( X Z ≤ lim TEN ); F ∈ R 3n is the vector of the forces, body forces and external forces, applied to the barycenter of the bricks;
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A T ∈ R3 n × m is the congruence operator, the transpose matrix of the equilibrium operator; x ∈R 3n is the displacement vector of the rigid body's barycenter; K ∈ R m × m is the compliance matrix of the rods; ∆ ∈R m is the vector of the inelastic strain which must be introduced in order to generate the suitable stress which will annul the tensile stresses unacceptable in the z rods simulating the mortar joints; vector ∆ contains the strains necessary to restablish congruence of the elastic strains stemming from the solution conforming to the sign constraint on the z rods. The strains contained in vector ∆ may be thought of as "cracks" due to the lack of tensile resistance in the rods modeling the mortar joints. Solution ( X , x ) and corresponding vector ∆ are determined through an iterative procedure in which: i) initially the elastic solution of the system ( X e , x e ) is determined by:
AX e + F = O AT xe + KX e = O
(2)
The elastic solution is then successively modified by acting on a single connecting rod until the solution satisfying condition X Z ≤ lim TEN is reached. Vector X o , corresponding to the first step, is obtained through:
X o = X e + CX N
(3)
C = I − K −1 AT ( AK −1 AT ) −1 A is the operator of orthogonal projection of R m on the null space N ( A) , and X N ∈ R m is an unknown vector determined by imposing the condition that within the set of stresses on the z connecting rods, stress X oj in rod j under maximum in which
tension major than bounded tensile strength lim TEN must be equal to zero:
X o j = X e j + CX Nj = 0
(4)
The equations for the problem thus become:
AX o + F = O A T x o + KX o + ∆ o = O
(5)
where ∆ 0 = KX n is the vector to be introduced in order to make the problem's kinematic equations congruent with the new solution ( X 0 = X e + CX N , x0 ) . ii) solution X i is successively modified by means of the correction
X i +1 = X i + CX N ( i +1)
(6)
in which C is the same as in the foregoing step and X N ( i +1) ∈ R m is an unknown vector determined by imposing the condition that the maximum stress from amongst those of the z rods with stress major than lim TEN and all those that have been annulled in previous steps be equal to zero:
X i +1 = X i + CX N ( i +1) = 0
(7)
At step (i + 1) solution ( X i +1 , xi +1 ) and the vector ∆ i +1 satisfy the equations
AX i +1 + F = O A T x i +1 + KX i+1 + ∆ i+1 = O
(8)
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where ∆ i +1 = KX N ( i +1) is the vector to be introduced in order to make the problem's kinematic equations congruent with the new solution ( X i +1 = X i + CX N (i +1) , xi +1 ) . The procedure ends when the stress is less than or equal to lim TEN for every z connecting rod. The Colonetti theorem guarantees that the linearly independent terms that can be introduced to modify the stress state in any given structure is equal, at most, to its degree of hyperstaticity. The algorithm has been implemented in Matlab (5.3R11). It is possible to input definitions for the geometry of the structure and the loads and stiffness of the connecting rods. The stiffness characterizes the various types of connecting rods (A,B and C in Fig. 1) and, in the case of type C (representing the reinforcement), the stiffness serves to model the width and thickness of the sheet. As output the algorithm furnishes the values of the forces in the connecting rods and the displacements of the blocks' barycenters, and displays the strained configuration and pressure curve corresponding to the applied loads.
3 EXPERIMENTAL TESTS Before performing tests on the 1:2 scale model masonry arches consolidated with FRP sheets, we conducted mechanical trials of the materials employed in constructing the models themselves, as well as tests of the adherence between the brick and FRP sheets. 3.1 Mechanical characteristics of the materials The mechanical characteristics of the bricks and mortar used to make the arch models were determined by monoaxial compression, direct tensile and bending stress tests. The compression tests were performed on prismatic brick specimens, 2.2 x 2.2 x 4.5 cm, made from brick blocks furnished by the firm “Laterizi S. Marco”, Venezia, Italia, as well as on 4 x 4 x 4 cm cubic specimens of cement-lime mortar whose mixture was constituted by one part hydrated "Fiore" type mortar, one part Portland composite cement (type II/a-L UNI ENV 197-1 R 32,5), eight parts Ticino sand and two parts water. Uniaxial compressive tests are performed also on 25 x 12 x 20 cm masonry samples. The direct tensile tests were conducted on 4 x 1 x 8 cm prismatic brick specimens with symmetrically arranged notches, approximately 1 cm in length. The bending stress tests were carried out on 11 x 25 x 5 cm prismatic brick samples and 4 x 4 x 16 cm prismatic lime-cement mortar samples. The mechanical parameter values determined by these tests are presented in Table I The "Mbrace Fibers C1-30" high-strength carbon fiber sheets, "Mbrace" primer and twocomponent epoxy-base adhesive that make up the MBrace FRP system were furnished by the firm MAC S.p.A. Treviso, Italia, who also provided the technical specifications shown in Table2. Table 1: Mechanical characteristics of the bricks, mortar and masonry employed for making the model arches. Specific Elastic Compressive Direct tensile Bending tensile weight modulus strength strength strength
Brick Mortar Masonry
[kg/m3]
(Mpa)
(Mpa)
(Mpa)
(Mpa)
1800 200 -
1785 133 830
17.39 7.8 8.6
1.7 -
3.53 2.26 -
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Table2: Technical and mechanical characteristics of the CFRP components (Mbrace system of MAC spa, Treviso, Italy). Lengthening at Bending Direct Bending Tensile Specific failure tensile tensile elastic elastic weight strength strength modulus modulus
Fiber C1-30 Primer Adhesive
[kg/l]
[Mpa]
1820 1067 1020
230000 >700 >3000
[Mpa]
[Mpa]
>580 >3500
>3430 >12 >50
[Mpa]
[%]
>24 >24
1.5 3 2.5
2.2. Adherence tests In order to check that the adhesive system of FRP Mbrace was also suitable for cladding the brick masonry support, we performed some direct tensile tests on specimens made up of two bricks connected end-to-end one to the other through a pair of fiber sheets of varying dimensions. Fifteen specimens were prepared, each made up of two parallelepiped 11 x 11 x 4 cm brick elements placed end to end with small faces abutting, and held together on both large faces by CFRP sheets of varying length and width. Obviously, with increasing surface area of the applied sheets, there is also a proportional increase in the sample's breaking load. Since detachment of the fiber sheets from the brick support never occurs, it can be stated that the adherence stress, calculated roughly as the ratio of breaking load to anchoring surface, results in all cases greater than 1 Mpa. 2.3. Tests on model arches Three masonry depressed 1:2 scale arches were prepared with double-layered brick and limecement mortar, with a span of 150 cm, rise of 43.25 cm and both height and thickness of 10 cm. These were subjected to the same conditions of increasing load up to collapse: arch n.1 without reinforcement and arches n.2 and n.3 with CFRP reinforcement sheets applied according to two different reinforcement arrangements. Arch n.2 were reinforced with 1.25 cm-wide CFRP sheet applied along the entire length of the intrados and arch n.3 were reinforced with 1.25 cm-wide CFRP sheet applied along the entire length of the extrados (Figure 3).
Figure 3: Arch n2 with CFRP sheet on intrados and arch n3 with CFRP sheet on extrados.
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The experimental trials (Figure 4) were performed at the Official Laboratory for Testing of Materials and Structures of the Construction Department of the University of Florence, Italy. The testing apparatus consisted of a rigid steel test frame that allowed imparting vertical displacement to the model's right springing, horizontal displacement to left springing and concentrated vertical loads at any point on the extrados. The load was applied gradually by means of a flywheel-operated screw jack able to impart increasing displacements, while the corresponding load increases were measured by means of a load cell with a capacity of 5KN and step of 50N, for the unreinforced arch, and a capacity of 100 kN and step of 100N for the consolidated arches.
Figure 4: The experimental trials.
Arc n.1 was brought to collapse in the absence of any reinforcement. The first stage of the loading process transformed the arches into isostatic structures through the formation of three hinges: one at the keystone on the extrados appeared at a load of 300N; and two others formed on the intrados near the haunches at a load of 450N. In the second stage of the loading process collapse ensued due to the emergence of two further hinges on the extrados in correspondence to the springings (Figure 5).
Figure 5: Arch n.1 at load of 300N.
Arches n.2 was damaged beforehand through a loading process that rendered them isostatic through the formation of three hinges, which occurred in correspondence to a load of 450N. After unloading, the arches were then consolidated through application of an CFRP sheet on the entire length of the intrados and were once again subjected to the nearly static application of loads at the key. Two hinges appeared on the intrados near the haunches in correspondence to a load of 1000N. Failure occurred due to detachment of the FRP sheet in correspondence to the
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keystone with a peak load of 3560N. After the almost complete detachment of the FRP sheet, the arch resumed the classical collapse mechanism by the emergence of three alternating hinges at the keystone and haunches, the same as that which occurred in the arches with no FRP-sheet reinforcement (Figure 6).
Figure 6 : Arch n.2 at load of 3560N.
Arches n.3 was damaged beforehand through a loading process that rendered them isostatic through the formation of three hinges, which occurred in correspondence to a load of 450N. After unloading, the arches were then consolidated through application of an FRP sheet on the entire length of the extrados and were once again subjected to the nearly static application of loads at the key. A hinge appeared on the extrados at the keystone in correspondence to a load of 750N. Subsequently two hinges appeared on the extrados near the springings in correspondence to a load of 2100N. Collapse ensued due to the emergence of two further hinges on the intrados near the haunches because of the local detachment of the FRP sheet. The collapse load was 3400N.
Figure 7: Arch n.3 at load of 750N.
load (N)
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4000 3500 3000 2500 2000 1500 1000
FRP on intrados FRP on extrados
500 0 0
1
2
3
4
5
6
displacement (mm)
Figure 8: Load-displacement diagrams recorded during the tests for arches n.2 e n.3.
Figure 8 shows the load-displacement diagrams recorded during the tests for the consolidation models. The load-displacement diagram reveals the followings characteristic points: the start of the linear segment (li), the end of the linear segment (l), the point of intersection between the extension of the linear segment and the point on the ordinate corresponding to the maximum load (l'), the maximum load (m) and the conventional ultimate load (u), which corresponds to 2/3 of the maximum. Using the values of the characteristic points recorded during the tests, the values of the following mechanical parameters were calculated: tangent stiffness Kt=(yl-yli)/(xl-xli), kinematic ductility µc=xm/xl’ and available kinematic ductility µcd=xm/xu. These have been summarized in Table 3, together with the values of the peak loads recorded. Table 3: mechanical parameters of the arches calculated using the values of the characteristic points recorded during the tests. Available ARCHES Peak load Tangent stiffness Kinematic ductility kinematic ductility (N) (N/µm) FRP on intrados 3560 0.24 2.2 1.1 FRP on extrados 3400 0.18 1.64 1.08
3 COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULTS Figures 9 and 10 show the numerical and experimental load-displacement diagrams for the arches n.2 and n.3. respectively: the parameters of numerical model come from the mechanical parameters of the materials employed to build the masonry arch models. The comparison between the experimental and numerical equilibrium pattern brings to the fore a very good fit, with the only exclusion of the values near the peak load. For these values of the load, the numerical model is stiffer than experimental. In correspondence of the peak load, the stiffness reduction of the experimental model is due to the damage of the masonry.
Historical Constructions
load (N)
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4000 3500 3000 2500 2000 1500 1000 experimental numerical
500 0 0
1
2
3
4
5
displacement (mm)
load (N)
Figure 9: numerical and experimental load-displacement diagrams for the arches n.2. 4000 3500 3000 2500 2000 1500 1000 500
experimental numerical
0 0
1
2
3
4
5
6
displacement (mm)
Figure 10: numerical and experimental load-displacement diagrams for the arches n.3.
REFERENCES Briccoli Bati S., Paradiso M., Tempesta G. 1998. Analysis of Masonry structures modelled by a set of rigid blocks and elastic unilateral contact constraints. In Pande G.N et al. (ed.), Computer Methods in Structural Masonry-4; Proc. Intern. Conf., Firenze. Briccoli Bati S., Rovero L. 2000. Consolidation of masonry arches with carbon-fiber reinforced plastics. In 12 TH International BRICK/BLOCK Masonry Conference, Madrid. Briccoli Bati S., Rovero L., 2000. Consolidation of masonry arches througt sheets of long carbon fibers composites, In Di Tommaso A. (ed.), Mechanics of masonry structures strengthened with FRP-materials; Proc. Nat. Conf., Venezia. Padova: Edizioni Libreria Cortina. Christensen J.B., Gilstrap J., Dolan C. 1996. Composites material Reinforcement of existing masonry walls. Journal of Architectural Engng vol 2 n2, p. 63- 70. Ehsani M.R., Saadatmanesh H., Velazquez-Dimas 1999. Behavior o Retrofitted URM walls under simulated earthquake loding. Jour of Composites for Construction vol 3, No 3, p. 134-142. Kolsch H., 1998. Carbon fiber cement overlay system for masonry strengthening. J. Composites for Construction, ASCE 2(5), p. 96-104. Kolsch H. 1998. Carbon Fiber Cement Matrix (CFCM) Overlay Sistem for Masonry Strengthening. Jour of Composites for Construction vol 2, No 2, p. 105-109. Luciano R., Sacco E. 1998. Strengthening of masonry panels with FRP composites. Int. J. Solids Structures 35(15), p.1723-1741. Nanni A. 1993. Fiber_Reinforced_Plastic Reinforcement for Concrete Structures:Properties and Applications. Amsterdam: Elsevier. Saadatmanesh, H., “Fiber composites for new and existing structures”, Structural J.,ACI, 91(3), pp.346354, 1993. Triantafillou T. C., Fardis M.N. 1997. Strengthening of hystoric masonry structures with composite materials. Materials and Structures RILEM, vol 30, n 202, p 486-496. Triantafillou T. C. 1998. Strengthening of masonry structures using epoxy-bonded FRP laminates. J. Composites for Construction, 2(2) ASCE, p.96-104.
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Experimental validation of a proposed numerical model for the FRP consolidation of masonry arches S. Briccoli Bati and L. Rovero University of Florence, Dipartimento di Costruzioni, Firenze, Italy
ABSTRACT: In planning the restoration and consolidation of historical buildings one must assess the effects on their structural behavior of applying advanced materials and technologies to elements made with traditional ones. The present paper deals with the consolidation of masonry arches through application of Carbon Fiber Reinforced Polymers (CFRP). A numerical model for arches reinforced with FRP sheets is proposed, and its validity checked through experimental trials. The numerical model represents the arch as a rigid assembly of blocks joined by elastic connecting rods with bounded tensile strength, which simulate the mortar joint, and tensionresistant connecting rods on the intrados and/or extrados, which simulate the FPR reinforcement. The numerical model has been checked experimentally through a series of tests on 1:2 scale models of brick arches subjected to the application of a load concentrated at the keystone.
1 INTRODUCTION Safeguarding the world's historical buildings requires defining appropriate methodologies for their restoration and consolidation. In defining such methodologies, one must account for, on the one hand, structural safety, including seismic hazard, and on the other, respect for the original architecture from both the aesthetic as well as structural perspectives. The consolidation techniques recommended and widely used in recent years for masonry structures include the addition of reinforced-concrete floors or centering to the extrados of vaulted structures. These, however, represent highly invasive and irreversible operations. In fact, apart from altering the original state and function of the structures under restoration, such methods often introduce extraneous elements that are incompatible with the pre-existing conventional materials and techniques, so much so that, at least in some cases, their application winds up contributing decisively to ruining the structure they were supposed to preserve. With regard to the many issues associated with historical building maintenance and restoration, the interest of workers in the sector has increasingly turned to the development of advanced technologies and the adoption of innovative materials. Many innovative materials are able to satisfy mechanical requirements not met by traditional materials and technologies. Of these, composite materials represent a particularly valuable group and have already been applied widely in aeronautics, mechanics and sporting equipment. The composite materials most often used in the fields of civil engineering and restoration are the FRP (Fiber Reinforced Polymer).
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FRP are composite material make by the union of fibers and polymeric matrix. The fibers are the strong components and are short and random arranged or long and in one or more direction arranged. The fibers are either glasses fibers (GFRP, Glasses Fiber Reinforced Polymer), either aramidic fibers (AFRP, Aramidic Fiber Reinforced Polymer), or carbon fibers (CFRP, Carbon Fiber Reinforced Polymer). The matrix protects the fibers, distributes the tensions and gives shape to the composite. The more used matrices are the thermosetting polymeric matrices. The principal characteristics of FRP are their high strength and stiffness relative to their weight, their considerable volumetric stability, good resistance to chemical action and low electrical conductivity. As they are anisotropic, they can be custom designed for specific tasks by arranging the fibers in such a way as to provide optimal load-bearing capacity for the particular loading conditions. The use of sheets of composite material in the form of cladding, applied to either the intrados or the extrados of vaulted structures, could obviate at least some of the inherent drawbacks of the consolidation techniques currently in use. In order to assess the efficiency of such reinforcement techniques, we conducted an experimental analysis on 1:2 scale models of brick arches consolidated by application of CFRP sheets and subjected to the application of a load concentrated at the keystone. The experimental results obtained were then utilized to check a numerical model that simulates the behavior of masonry arches reinforced with FRP cladding. The arch is modeled as a series of rigid blocks joined at their interfaces by elastic connecting rods. In particular the voussoir is modeled as rigid block; the mortar-joint is modeled as a series of elastic rods with bounded tensile strength; the FRP reinforcement, in correspondence to the intrados and extrados, is simulated by elastic tension-resistant connecting rods endowed with suitable extensional stiffness.
2. NUMERICAL MODEL
2.1 Description In Briccoli Bati et al. (1998) a numerical model is introduced for calculations on masonry arches, under the assumptions that the voussoirs are rigid and the interfaces elastic-cracking. Such modeling furnishes results in perfect agreement with experimental results and able to accurately interpret actual cases. In the present work the numerical model has been modified in order to enable its application to calculations relative to arches reinforced with FRP sheets placed on the intrados and/or extrados.
A
RIGID BLOCK
B
ELASTIC NO-TENSION ROD
C
ELASTIC ROD
D
ELASTIC ROD
Figure 1: Numerical model.
S. B. Bati and L. Rovero
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The proposed model conceives of the reinforced arch as made up of rigid blocks fixed to the interface by three types of appropriately rigid elastic connecting rods. In Figure 1 the connecting rods indicated by the letters A and B simulate the mortar joints: more precisely, type A joints, tangent to the arch axis, have bounded tensile C and absorb the normal component, while type B, orthogonal to the arch axis, resists both tension and compression and absorbs the shear component. Connecting rods C, resistant alone to tension, are set on the intrados or extrados and represent the FRP reinforcement. The rods A with bonded tensile strength were eliminated, when the internal force is major or equal to ultimate tensile strength. The elimination of the rods fits the reduction of section because there is a fracture due to the no-tension strength of the masonry. The stiffness of the rod B corresponding to eliminated rods A is reduced in proportion to the number of eliminated rods A in the section. The mechanical parameters of the model are the elastic modulus of the rods A and B, the ultimate tensile strength of the rods A and the elastic modulus of the rods C. The elastic modulus and the tension limit of A rods depend on mechanical characteristics of joint mortar. The elastic modulus of C rods depend on mechanical characteristics of the FRP reinforcement. The elastic modulus of the rods B depends on the mechanical characteristics of joint mortar and on geometric characteristics of the section by means of following formula:
Nl B λTt = GAsez ErodB ArodB with λ shear factor, T tangential force, t joint thickness, G elastic tangential modulus of the mortar, As e z surface of the cross section of the structure, N normal force, lB length of rod B equal to height of structure section, ArodB surface of the cross section of the rod B. E rodB is determined imposing that the longitudinal deformation of the rod B with normal force is equal to the deformation produced by the tangential force T (equal to normal force N ) in the section of the structure. 2.2 The no-tension problem’s equations The constitutive model for a stucture consisted of rigid blocks and elastic roods with bounded tensile streght is derived from linear elastic model through the correction of the static solution and through the following determination of the kinematic solution consistent with the new static solution. The static solution is determined adding selfequilibrated stress to elastic stress to annul the not acceptable tensile tress. The kinematic solution is determined considering the selfequilibrated stress as internal forces resulted from applied strains that give validity to kinematic equations. The applied strains are as fractures caused of the bounded tensile strenght. In particular the equation of the problems are: for a structure made up of n rigid bodies connected by m elastic connecting rods (fig.2), of which z with bounded tensile strength lim TEN , the equations for the static and kinematic problem are AX + F = O A T x + KX + ∆ = O
(1)
X Z ≤ lim TEN
where: A ∈ R m×3 n is the equilibrium operator or topological matrix; X ∈R m is the vector of the normal forces in the connecting rods; z are the rods absorbing the normal component of the mortar joints and therefore with bounded tensile strength ( X Z ≤ lim TEN ); F ∈ R 3n is the vector of the forces, body forces and external forces, applied to the barycenter of the bricks;
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A T ∈ R3 n × m is the congruence operator, the transpose matrix of the equilibrium operator; x ∈R 3n is the displacement vector of the rigid body's barycenter; K ∈ R m × m is the compliance matrix of the rods; ∆ ∈R m is the vector of the inelastic strain which must be introduced in order to generate the suitable stress which will annul the tensile stresses unacceptable in the z rods simulating the mortar joints; vector ∆ contains the strains necessary to restablish congruence of the elastic strains stemming from the solution conforming to the sign constraint on the z rods. The strains contained in vector ∆ may be thought of as "cracks" due to the lack of tensile resistance in the rods modeling the mortar joints. Solution ( X , x ) and corresponding vector ∆ are determined through an iterative procedure in which: i) initially the elastic solution of the system ( X e , x e ) is determined by:
AX e + F = O AT xe + KX e = O
(2)
The elastic solution is then successively modified by acting on a single connecting rod until the solution satisfying condition X Z ≤ lim TEN is reached. Vector X o , corresponding to the first step, is obtained through:
X o = X e + CX N
(3)
C = I − K −1 AT ( AK −1 AT ) −1 A is the operator of orthogonal projection of R m on the null space N ( A) , and X N ∈ R m is an unknown vector determined by imposing the condition that within the set of stresses on the z connecting rods, stress X oj in rod j under maximum in which
tension major than bounded tensile strength lim TEN must be equal to zero:
X o j = X e j + CX Nj = 0
(4)
The equations for the problem thus become:
AX o + F = O A T x o + KX o + ∆ o = O
(5)
where ∆ 0 = KX n is the vector to be introduced in order to make the problem's kinematic equations congruent with the new solution ( X 0 = X e + CX N , x0 ) . ii) solution X i is successively modified by means of the correction
X i +1 = X i + CX N ( i +1)
(6)
in which C is the same as in the foregoing step and X N ( i +1) ∈ R m is an unknown vector determined by imposing the condition that the maximum stress from amongst those of the z rods with stress major than lim TEN and all those that have been annulled in previous steps be equal to zero:
X i +1 = X i + CX N ( i +1) = 0
(7)
At step (i + 1) solution ( X i +1 , xi +1 ) and the vector ∆ i +1 satisfy the equations
AX i +1 + F = O A T x i +1 + KX i+1 + ∆ i+1 = O
(8)
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where ∆ i +1 = KX N ( i +1) is the vector to be introduced in order to make the problem's kinematic equations congruent with the new solution ( X i +1 = X i + CX N (i +1) , xi +1 ) . The procedure ends when the stress is less than or equal to lim TEN for every z connecting rod. The Colonetti theorem guarantees that the linearly independent terms that can be introduced to modify the stress state in any given structure is equal, at most, to its degree of hyperstaticity. The algorithm has been implemented in Matlab (5.3R11). It is possible to input definitions for the geometry of the structure and the loads and stiffness of the connecting rods. The stiffness characterizes the various types of connecting rods (A,B and C in Fig. 1) and, in the case of type C (representing the reinforcement), the stiffness serves to model the width and thickness of the sheet. As output the algorithm furnishes the values of the forces in the connecting rods and the displacements of the blocks' barycenters, and displays the strained configuration and pressure curve corresponding to the applied loads.
3 EXPERIMENTAL TESTS Before performing tests on the 1:2 scale model masonry arches consolidated with FRP sheets, we conducted mechanical trials of the materials employed in constructing the models themselves, as well as tests of the adherence between the brick and FRP sheets. 3.1 Mechanical characteristics of the materials The mechanical characteristics of the bricks and mortar used to make the arch models were determined by monoaxial compression, direct tensile and bending stress tests. The compression tests were performed on prismatic brick specimens, 2.2 x 2.2 x 4.5 cm, made from brick blocks furnished by the firm “Laterizi S. Marco”, Venezia, Italia, as well as on 4 x 4 x 4 cm cubic specimens of cement-lime mortar whose mixture was constituted by one part hydrated "Fiore" type mortar, one part Portland composite cement (type II/a-L UNI ENV 197-1 R 32,5), eight parts Ticino sand and two parts water. Uniaxial compressive tests are performed also on 25 x 12 x 20 cm masonry samples. The direct tensile tests were conducted on 4 x 1 x 8 cm prismatic brick specimens with symmetrically arranged notches, approximately 1 cm in length. The bending stress tests were carried out on 11 x 25 x 5 cm prismatic brick samples and 4 x 4 x 16 cm prismatic lime-cement mortar samples. The mechanical parameter values determined by these tests are presented in Table I The "Mbrace Fibers C1-30" high-strength carbon fiber sheets, "Mbrace" primer and twocomponent epoxy-base adhesive that make up the MBrace FRP system were furnished by the firm MAC S.p.A. Treviso, Italia, who also provided the technical specifications shown in Table2. Table 1: Mechanical characteristics of the bricks, mortar and masonry employed for making the model arches. Specific Elastic Compressive Direct tensile Bending tensile weight modulus strength strength strength
Brick Mortar Masonry
[kg/m3]
(Mpa)
(Mpa)
(Mpa)
(Mpa)
1800 200 -
1785 133 830
17.39 7.8 8.6
1.7 -
3.53 2.26 -
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Table2: Technical and mechanical characteristics of the CFRP components (Mbrace system of MAC spa, Treviso, Italy). Lengthening at Bending Direct Bending Tensile Specific failure tensile tensile elastic elastic weight strength strength modulus modulus
Fiber C1-30 Primer Adhesive
[kg/l]
[Mpa]
1820 1067 1020
230000 >700 >3000
[Mpa]
[Mpa]
>580 >3500
>3430 >12 >50
[Mpa]
[%]
>24 >24
1.5 3 2.5
2.2. Adherence tests In order to check that the adhesive system of FRP Mbrace was also suitable for cladding the brick masonry support, we performed some direct tensile tests on specimens made up of two bricks connected end-to-end one to the other through a pair of fiber sheets of varying dimensions. Fifteen specimens were prepared, each made up of two parallelepiped 11 x 11 x 4 cm brick elements placed end to end with small faces abutting, and held together on both large faces by CFRP sheets of varying length and width. Obviously, with increasing surface area of the applied sheets, there is also a proportional increase in the sample's breaking load. Since detachment of the fiber sheets from the brick support never occurs, it can be stated that the adherence stress, calculated roughly as the ratio of breaking load to anchoring surface, results in all cases greater than 1 Mpa. 2.3. Tests on model arches Three masonry depressed 1:2 scale arches were prepared with double-layered brick and limecement mortar, with a span of 150 cm, rise of 43.25 cm and both height and thickness of 10 cm. These were subjected to the same conditions of increasing load up to collapse: arch n.1 without reinforcement and arches n.2 and n.3 with CFRP reinforcement sheets applied according to two different reinforcement arrangements. Arch n.2 were reinforced with 1.25 cm-wide CFRP sheet applied along the entire length of the intrados and arch n.3 were reinforced with 1.25 cm-wide CFRP sheet applied along the entire length of the extrados (Figure 3).
Figure 3: Arch n2 with CFRP sheet on intrados and arch n3 with CFRP sheet on extrados.
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The experimental trials (Figure 4) were performed at the Official Laboratory for Testing of Materials and Structures of the Construction Department of the University of Florence, Italy. The testing apparatus consisted of a rigid steel test frame that allowed imparting vertical displacement to the model's right springing, horizontal displacement to left springing and concentrated vertical loads at any point on the extrados. The load was applied gradually by means of a flywheel-operated screw jack able to impart increasing displacements, while the corresponding load increases were measured by means of a load cell with a capacity of 5KN and step of 50N, for the unreinforced arch, and a capacity of 100 kN and step of 100N for the consolidated arches.
Figure 4: The experimental trials.
Arc n.1 was brought to collapse in the absence of any reinforcement. The first stage of the loading process transformed the arches into isostatic structures through the formation of three hinges: one at the keystone on the extrados appeared at a load of 300N; and two others formed on the intrados near the haunches at a load of 450N. In the second stage of the loading process collapse ensued due to the emergence of two further hinges on the extrados in correspondence to the springings (Figure 5).
Figure 5: Arch n.1 at load of 300N.
Arches n.2 was damaged beforehand through a loading process that rendered them isostatic through the formation of three hinges, which occurred in correspondence to a load of 450N. After unloading, the arches were then consolidated through application of an CFRP sheet on the entire length of the intrados and were once again subjected to the nearly static application of loads at the key. Two hinges appeared on the intrados near the haunches in correspondence to a load of 1000N. Failure occurred due to detachment of the FRP sheet in correspondence to the
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keystone with a peak load of 3560N. After the almost complete detachment of the FRP sheet, the arch resumed the classical collapse mechanism by the emergence of three alternating hinges at the keystone and haunches, the same as that which occurred in the arches with no FRP-sheet reinforcement (Figure 6).
Figure 6 : Arch n.2 at load of 3560N.
Arches n.3 was damaged beforehand through a loading process that rendered them isostatic through the formation of three hinges, which occurred in correspondence to a load of 450N. After unloading, the arches were then consolidated through application of an FRP sheet on the entire length of the extrados and were once again subjected to the nearly static application of loads at the key. A hinge appeared on the extrados at the keystone in correspondence to a load of 750N. Subsequently two hinges appeared on the extrados near the springings in correspondence to a load of 2100N. Collapse ensued due to the emergence of two further hinges on the intrados near the haunches because of the local detachment of the FRP sheet. The collapse load was 3400N.
Figure 7: Arch n.3 at load of 750N.
load (N)
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4000 3500 3000 2500 2000 1500 1000
FRP on intrados FRP on extrados
500 0 0
1
2
3
4
5
6
displacement (mm)
Figure 8: Load-displacement diagrams recorded during the tests for arches n.2 e n.3.
Figure 8 shows the load-displacement diagrams recorded during the tests for the consolidation models. The load-displacement diagram reveals the followings characteristic points: the start of the linear segment (li), the end of the linear segment (l), the point of intersection between the extension of the linear segment and the point on the ordinate corresponding to the maximum load (l'), the maximum load (m) and the conventional ultimate load (u), which corresponds to 2/3 of the maximum. Using the values of the characteristic points recorded during the tests, the values of the following mechanical parameters were calculated: tangent stiffness Kt=(yl-yli)/(xl-xli), kinematic ductility µc=xm/xl’ and available kinematic ductility µcd=xm/xu. These have been summarized in Table 3, together with the values of the peak loads recorded. Table 3: mechanical parameters of the arches calculated using the values of the characteristic points recorded during the tests. Available ARCHES Peak load Tangent stiffness Kinematic ductility kinematic ductility (N) (N/µm) FRP on intrados 3560 0.24 2.2 1.1 FRP on extrados 3400 0.18 1.64 1.08
3 COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULTS Figures 9 and 10 show the numerical and experimental load-displacement diagrams for the arches n.2 and n.3. respectively: the parameters of numerical model come from the mechanical parameters of the materials employed to build the masonry arch models. The comparison between the experimental and numerical equilibrium pattern brings to the fore a very good fit, with the only exclusion of the values near the peak load. For these values of the load, the numerical model is stiffer than experimental. In correspondence of the peak load, the stiffness reduction of the experimental model is due to the damage of the masonry.
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load (N)
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4000 3500 3000 2500 2000 1500 1000 experimental numerical
500 0 0
1
2
3
4
5
displacement (mm)
load (N)
Figure 9: numerical and experimental load-displacement diagrams for the arches n.2. 4000 3500 3000 2500 2000 1500 1000 500
experimental numerical
0 0
1
2
3
4
5
6
displacement (mm)
Figure 10: numerical and experimental load-displacement diagrams for the arches n.3.
REFERENCES Briccoli Bati S., Paradiso M., Tempesta G. 1998. Analysis of Masonry structures modelled by a set of rigid blocks and elastic unilateral contact constraints. In Pande G.N et al. (ed.), Computer Methods in Structural Masonry-4; Proc. Intern. Conf., Firenze. Briccoli Bati S., Rovero L. 2000. Consolidation of masonry arches with carbon-fiber reinforced plastics. In 12 TH International BRICK/BLOCK Masonry Conference, Madrid. Briccoli Bati S., Rovero L., 2000. Consolidation of masonry arches througt sheets of long carbon fibers composites, In Di Tommaso A. (ed.), Mechanics of masonry structures strengthened with FRP-materials; Proc. Nat. Conf., Venezia. Padova: Edizioni Libreria Cortina. Christensen J.B., Gilstrap J., Dolan C. 1996. Composites material Reinforcement of existing masonry walls. Journal of Architectural Engng vol 2 n2, p. 63- 70. Ehsani M.R., Saadatmanesh H., Velazquez-Dimas 1999. Behavior o Retrofitted URM walls under simulated earthquake loding. Jour of Composites for Construction vol 3, No 3, p. 134-142. Kolsch H., 1998. Carbon fiber cement overlay system for masonry strengthening. J. Composites for Construction, ASCE 2(5), p. 96-104. Kolsch H. 1998. Carbon Fiber Cement Matrix (CFCM) Overlay Sistem for Masonry Strengthening. Jour of Composites for Construction vol 2, No 2, p. 105-109. Luciano R., Sacco E. 1998. Strengthening of masonry panels with FRP composites. Int. J. Solids Structures 35(15), p.1723-1741. Nanni A. 1993. Fiber_Reinforced_Plastic Reinforcement for Concrete Structures:Properties and Applications. Amsterdam: Elsevier. Saadatmanesh, H., “Fiber composites for new and existing structures”, Structural J.,ACI, 91(3), pp.346354, 1993. Triantafillou T. C., Fardis M.N. 1997. Strengthening of hystoric masonry structures with composite materials. Materials and Structures RILEM, vol 30, n 202, p 486-496. Triantafillou T. C. 1998. Strengthening of masonry structures using epoxy-bonded FRP laminates. J. Composites for Construction, 2(2) ASCE, p.96-104.
