Historical Constructions, P.B. Lourenço, P. Roca (Eds.), Guimarães, 2001 539

1 INTRODUCTION

Brick masonry is an ancient building technique, and masonry structures constitute a large portion of buildings around the world. Nowadays, preservation of the architectural heritage is considered a fundamental issue and the research interest in this area has begun to increase.

Masonry is a composite material made of brick units, usually made from clay, and mortar joints. The large number of variables influencing the mechanical behavior of masonry, e.g. mate-rial properties of brick and mortar, geometry of bricks, joint dimensions, joint arrangement, etc. forced the early analyses to be dramatically simple; masonry was often assumed to be isotropic elastic.

In the last years, some more refined models are adopted to take in account the nonlinear effects due to the fracture openings and damage of the material.

In particular, the damage theory model revealed to be a good choice, especially because of its efficiency combined with simplicity, Creazza et al. (2000).

Because of the low tensile strength of the masonry material, structural engineers have coupled masonry with several kinds of reinforcements to overcome such intrinsic weakness. The most used materials for the reinforcements of masonry are concrete and iron, but nowadays new mate-rials, such as composites, begin to be used.

Composite materials have been successfully used in several fields of structural engineering, mainly in Aerospace and Mechanical Engineering. Nowadays, fiber-reinforced plastic (FRP) is adopted to replace, or complement, traditional materials also in Civil Engineering. In fact, FRP materials are characterised by several advantages, such as low weight, high strength and also good resistance to corrosion and durability, (Third Int. Conf. 2000).

Beside, the FRP satisfy the modern requirements for a modern intervention on restoration, like minimum repair and respect of the original construction and reversibility. So, although the FRP material is more expensive than many traditional materials, its use can become advantageous both in the construction of new building and in the restoration of ancient building.

Several techniques for the strengthening of masonry structures with advanced composites have been proposed but, recently, two kinds of reinforcements are emerging: i.e. embedding of rods

Analysis of masonry structures reinforced by FRP

Giuseppe Creazza, Anna V. Saetta IUAV - Department of Architectural Construction, Venice, Italy

Renato Matteazzi, Renato V. Vitaliani University of Padova, Department of Construction and Transportation, Padova Italy

ABSTRACT: The analysis of masonry vaults reinforced by FRP materials has been carried out by means of two-parameters damage model within the framework of three-dimensional analyses. Masonry is treated as a homogenized material, for which the material characteristics can be de-fined by using homogenization technique as well as micro-modeling approach, while the FRP is simulated by elastic constitutive law. A cylindrical vault, which has been experimentally tested, is studied: the analysis is performed on the vault by considering the presence of FRP both in the in-tradox and extradox. The agreement of numerical results with experimental data is such that the accuracy and the effectiveness of the damage approach could be assert in the analysis of large real structures.

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into grooves close to the surface (near-surface mounted FRP rods) and bonding of composites laminates to the facades.

Experimental tests on masonry elements have been carried out to evaluate the effectiveness of both strengthening techniques, (De Lorenzis et al. 2000, Tinazzi et al. 2000, Faccio and Forabo-schi 2000a) and also some numerical applications have been developed to simulate the behaviour of the reinforced masonry (e.g. Luciano and Sacco 1998, 3 rd Inter. Conf. 2000).

In the present work, the mechanical behaviour of some masonry structures reinforced by FRP is numerically analysed. A two-parameters, scalar, isotropic, damage model is used to represent the masonry, considered as an homogeneous material, and an elastic constitutive law for the FRP material is adopted.

The capabilities and the validity limits of the finite element analysis obtained by the continuum approach are presented and discussed, with regard to its ability to simulate the mechanical behav-iour exhibited in an experimental test carried out on masonry vaulted structure reinforced by FRP.

2 TWO-PARAMETER DAMAGE APPROACH

2.1 Basic assumptions

A scalar damage model with two different internal damage variables, d+ and d-, respectively for tensile and compressive stresses contribution, has been adopted to simulate masonry behavior.

The related expression for the Cauchy stress tensor assumes the following form (e.g. Saetta et al. 1998, 1999):

( ) ( ) ( ) ( ) −−++−

−+

+ σ−+σ−=∂εΨ∂

−+∂εΨ∂

−=∂ε

Ψ∂=σ ijije

ij

0eij

0eij

ij d1d1d1d1 (1)

where +Ψ0 and −Ψ0 are elastic free energies related to the tensile and compressive components of

the effective stress tensor, which is splitted into two components +σij and −σij . Such a damage model has been successfully used to describe the behavior of r.c. structures,

(e.g. Saetta et al. 1998 and 1999) and masonry constructions (Oñate et al. 1997, Creazza et al. 1995 and 2000) under different load conditions.

Figure 1 shows the constitutive law used for the masonry in the three-dimensional finite ele-ment analysis. It has been obtained from the experimental tests carried out on some masonry specimens. Similarly Figure 2 shows the related damage vs strain diagram.

compression

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

-0.020 -0.015 -0.010 -0.005 0.000

Strain

Str

ess

(MP

a)

traction

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025Strain

Str

ess

(MP

a)

Figure 1: Masonry constitutive law

G. Creazza, A. V. Saetta, R. Matteazzi and R. V. Vitaliani 541

-0.25

0

0.25

0.5

0.75

1

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005

strain

dam

age

Figure 2: Damage parameters vs strain

3 REINFORCED CYLINDRICAL VAULT

3.1 Geometry and experimental set-up

The cylindrical vault is subjected to the action of its own weight and a point load P, near quarter-span. The vault thickness is of 130 mm and the span of 2000 mm.

Three continuous FRP reinforcements sheets have been bonded along the intrados of the vault (Figure 3a). They have 64 cm spacing and each sheet is 5 cm large and 1.2 mm thick.

(a) (b)

Figure 3: (a) disposition of the FRP at the intrados of the cylindrical vault (b) particular of the vault: the debonding of FRP

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The vault has been experimentally tested and during the test, the single point load P is slowly increased, so that its effect becomes more and more important compared with that of the dead weight of the vault.

As the point load P increases, the subsequent formation of four cracked zones has been ob-served, which would correspond to the opening of four hinges (i.e. the classical four-bar chains mechanism), but the presence of reinforcements avoided the formation of the classical unstable mechanism of collapse.

At a certain value of point load P, experimentally find equal to 36 kN, the vault can no longer contain the structural forces and an unstable mechanism of collapse formed.

The vertical displacement at the midspan (channel 1, see Figure 5) under such a limit load was about 4.77 mm. The experimental failure pattern for the

cylindrical vault is shown in Figure 4, where it is also evidenced the sequence of formation of the four hinges.

Figure 4: Experimental failure pattern for the reinforced cylindrical vault (Faccio, Foraboschi 2000b)

3.2 Numerical analysis

The experimental test on the reinforced cylindrical vault described in the previous section has simulated by using both two- and the three-dimensional damage model.

Figure 5 shows the bidimensional and the three-dimensional finite element meshes used in the numerical analyses, where the positions of the experimental set-up (i.e. the channels for the dis-placements recording and the position of the applied load) is also indicated.

The values chosen for the material parameters of masonry are summarized in Table 1, while the FRP reinforcements are described by elements with elastic constitutive law, with Young’s modulus equal to 165000 MPa.

3rd hinge

4th hinge

2nd hinge

1st hinge

G. Creazza, A. V. Saetta, R. Matteazzi and R. V. Vitaliani 543

Table 1: Material properties of masonry for the 3-D cylindrical vault undamaged elastic modulus Ec = 1700 MPa

Poisson’s ratio ν = 0.25

uniaxial (1D) initial compressive strength fc = 3.0 MPa

2D strength/1D strength fc1D/fc2D = 1.2

Linear limit 0.6 fc

uniaxial (1D) initial tensile strength ft = 0.1 MPa

Fracture Energy Gf = 0.0312 N/mm

(a) (b)

Figure 5: (a) 2-D finite element mesh; (b) 3-D finite element mesh

The load-displacement diagrams obtained both with the 2D and the 3D finite element analyses

are depicted in Figure 6, compared with the experimental one. The peak load of 35 kN is reached for a displacement equal to 5 mm, in good agreement with

the experimental data.

0

5000

10000

15000

20000

25000

30000

35000

40000

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

displacement (mm)

load

(N)

Figure 6: load-displacement diagram for the cylindrical vault

experimental

2-D

3-D

ch 2 ch 1

ch 2 ch 1

ch 1 ch 2

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Figure 7 shows the sequence of d+ damage contours for four different steps of the analysis, cor-responding to the formation of the four hinges, in perfect agreement with the experimental evi-dence.

It is worth noting that in correspondence of the formation of the fourth damaged zones, at a displacement of 1.16 mm (very far from the collapse displacement of 5 mm) the vault maintains a load carrying capacity, due to the presence of FRP reinforcements.

Figure 8 shows the 3-D deformed configuration of the cylindrical vault in correspondence of the collapse load.

(a) (b)

(c) (d)

Figure 7: d+ damage contour at a displacement of:

: (a) 0.4 mm; (b) 0.45 mm; (c) 0.71 mm; (d) 1.16 mm

Figure 8: Deformed configuration of 3-D model at collapse

G. Creazza, A. V. Saetta, R. Matteazzi and R. V. Vitaliani 545

4 CONCLUSIONS

The proposed damage model allows analysing the global structural response of masonry struc-tures, both without and with FRP reinforcements. In the present version of the proposed damage approach, the modelling of FRP is very simplified and a further improvement of such a research will be the development of suitable constitutive law and finite element to better simulate the cou-pling of masonry and FRP.

However, till in this version, the model could be considered sufficiently accurate to understand the global behaviour of reinforced masonry and, in particular, to capture the changes of failure mechanism and the increase of the carrying-load capacity due to the FRP introduction.

REFERENCES

Creazza G., Saetta A., Scotta R., Vitaliani R., Oñate E. 1995. Mathematical simulation of structural damage in historical buildings. In Brebbia and Leftheris (ed.) Architectural Studies, Materials & Analysis, Proc, STREMA 95, Structural Studies of Historical Buildings, Crete, Greece, May 22-24, 1995, Comp Mech. Publ., pp. 111-118, vol 1

Creazza G., Matteazzi R., Saetta A., Vitaliani R. 2000. Analyses of masonry vaulted structures by using 3-D damage model, In European Congress on Computational Methods in Applied Sciences and En-gineering, ECCOMAS 2000, Barcelona, 11-14 September 2000

De Lorenzis, L., D. Tinazzi, A. Nanni, A. 2000. Near Surface Mounted FRP Rods for Masonry Strengthening: Bond and Flexural Testing, In Symposium, “Meccanica delle Strutture in Muratura Rinforzate con FRP Materials,” Venezia, Italy, December 7-8, 2000

Di Marco R., Faccio P., Foraboschi P., Siviero E. 1999, Volte in muratura rinforzate con FRP”, Costruire in laterizio, 66-71.

Faccio P., Foraboschi P., Siviero E. 1999. Volte in muratura con rinforzi in FRP, L’Edilizia, n. 9/10, 44-59.

Faccio P., Foraboschi P. 2000(a). Experimental and theoretical analysis of masonry vaults with FRP re-inforcements, In 3rd Int. Conference on advanced composite materials in bridges and structures, Ot-tawa, Canada.

Faccio P., Foraboschi P. 2000(b). Analisi Limite Ultima di Strutture in Muratura con Materiali Compo-siti Incollati al Contorno, Atti del Convegno Nazionale Meccanica delle strutture in muratura rinfor-zate con FRP – materials, Venezia 7-8 Dec. 2000

Heyman J. 1996. Arches, vaults and buttresses: masonry structures and their engineering, Variorum Ashgate Publishing.

Luciano R., Sacco E. 1998. Damage of masonry panels reinforced by FRP sheets, Int. J. Solids Struc-tures, Vol. 35, N° 15, pp. 1723-1741.

Oñate E., Hanganu A., Barbat A., Oller S., Vitaliani R., Saetta A., Scotta R. 1997. Structural analysis and durability assessment of historical construction using a finite element damage model. In P. Roca, J.L. Gonzàlez, A.R. Marì and Oñate E. (ed.), Structural Analysis of Historical construction, pp. 189-224, Barcelona: CIMNE.

Saetta A., Scotta R., Vitaliani R. 1998. Seismic Analysis of Reinforced Concrete Frames with a Scalar Damage Model. In E. Oñate and S. R. Idelsohn (Eds.) Computational Mechanics, New Trends and Applications, Proc. Fourth WCCM Congress, Buenos Aires 29 June - 2 July, 1998, Barcelona: CIMNE.

Saetta A., Scotta R., Vitaliani R. 1999. Coupled Environmental-Mechanical Damage Model of RC Structures, Journal of Engineering Mechanics, ASCE, pp. 930-940, August 1999, vol. 125, issue 8

Third Inter. Conf. on Advanced Composite Materials in Bridges and Structures, Ottawa, Canada, J. Humar and A.G. Razaqpur, Editors, 15-18 Aug. 2000

Tinazzi, D., Modena C., and Nanni A. 2000. Strengthening of Masonry Assemblages with FRP Rods and Laminates, In Crivelli-Visconti (eds), International Meeting on Composite Materials, PLAST 2000, Proceedings, Advancing with Composites 2000, Milan, Italy, May 9-11, 2000, pp. 411-418.

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