Preface

The primary intention of this book is to embark on a journey that takes the reader through the latest concepts, techniques, and trends in structural engineering. This publication is mainly focused on structural engineering principles in the simulation, modeling, and design of structural elements. In addition, factors affecting structural response and behavior are in‐ vestigated. The book is addressed to researchers and practitioners on the subject, as well as undergraduate and postgraduate students and other experts and newcomers seeking more information about the state of the art, new challenges, innovative solutions, and new trends and developments in these areas.

The behavior and design of masonry walls is the subject of Chapters 1 and 2. In the first chapter, two formulations were proposed by the authors for masonry walls confined with fiber-reinforced polymers, which are adjusted for the cohesion and internal friction angle of both units and mortar. In the second chapter, a discussion on experimental tests carried out on different types of fuse elements as well as quarter-scale specimen of frame and infill walls with fuse elements is presented. Later on, the results of finite element computer mod‐ eling studies are presented and verified through experimental analysis.

Chapter 3 is dedicated to a novel prefabricated steel-reinforced concrete composite column. In the proposed column, four steel angles are placed at the corners of the cross-section, and transverse bars and plates are used to connect the angles by welding or bolting. In this chap‐ ter, the newly developed composite column is introduced, and basic mechanisms, structural performance, and field applications are discussed.

Chapter 4 deals with the behavior of a simply-supported beam (SSB) reinforced with exter‐ nal prestressing using three diagonal web members. A theoretical model was created for the reinforced SSB. Then equations governing the relationship between beam deflection and in‐ crement in external tendon force and the relationship between load and deflection were de‐ rived. Later, finite element analysis was performed to study the effects of cross-sectional area of the tendon, tendon sag, initial tendon force, and other factors on the structural per‐ formance of an SSB after reinforcement.

The final chapter, Chapter 5, discusses the vertical natural vibration modes of a ballasted railway track. This study measured the natural vibration characteristics of the ballast layer using field measurements, full-scale impact loading experiments, and large-scale finite ele‐ ment analysis.

It is our hope that this book, which integrates the modeling and design aspects of various structural systems, will serve as a comprehensive guide and reference for practicing engi‐ neers and educators, and, more importantly, as a welcome mat for recent graduates entering the structural engineering profession by assuring them that they have discovered an excit‐ ing world of challenges and opportunities.

#### **Associate Professor Dr. Hakan Yalçiner**

**Section 1**

**Behavior and Design of Masonry Walls**

Department of Civil Engineering Erzincan University Erzincan, Turkey **Assistant Professor Dr. Ehsan Noroozinejad Farsangi** Department of Earthquake Engineering KGUT Iran **Behavior and Design of Masonry Walls**

It is our hope that this book, which integrates the modeling and design aspects of various structural systems, will serve as a comprehensive guide and reference for practicing engi‐ neers and educators, and, more importantly, as a welcome mat for recent graduates entering the structural engineering profession by assuring them that they have discovered an excit‐

**Associate Professor Dr. Hakan Yalçiner**

Department of Earthquake Engineering

**Assistant Professor Dr. Ehsan Noroozinejad Farsangi**

Department of Civil Engineering

Erzincan University Erzincan, Turkey

> KGUT Iran

ing world of challenges and opportunities.

VIII Preface

**Chapter 1**

**Provisional chapter**

**Constitutive Modeling of Masonry Walls Strengthened**

**Constitutive Modeling of Masonry Walls Strengthened** 

This study primarily focuses on the constitutive modeling of masonry walls strengthened with fiber-reinforced polymers (FRP). In general, providing suitable relations for the material characterization of the masonry constituents so that the nonlinear finite element (NLFEA) applications of the elasto-plastic theory achieves a close fit to the experimental load-displacement diagrams of the walls subjected to in-plane loadings is desired. In this chapter, two relations proposed for masonry columns confined with FRP are adjusted for the cohesion and the internal friction angle of both units and mortar. Relating the mechanical parameters to the uniaxial compression strength and the hydrostatic pressure acting over the wall surface, the effects of major and intermediate principal stresses σ<sup>1</sup> and σ<sup>2</sup> on the yielding and the shape of the deviatoric section are then reflected into the analyses. Finally, NLFEA results for the walls tested in different studies showed a good

**Keywords:** masonry wall, fiber reinforced polymer, cohesion, internal friction,

Many historical masonry buildings of the world's architectural heritage are in dire need of retrofitting as they do not perform well to the seismic activities. Retrofitting of such structures needed better understanding of the constitutive features to characterize its present condition and determine the true structural safety for seismic actions. Although masonry has been used in building for centuries in many countries such as Turkey, constitutive

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

DOI: 10.5772/intechopen.77201

**with Fiber Reinforced Polymers**

**with Fiber Reinforced Polymers**

Oktay Jafarov and Cengiz Karakoç

Oktay Jafarov and Cengiz Karakoç

http://dx.doi.org/10.5772/intechopen.77201

match with the experimental ones.

elasto-plastic analysis, finite element analysis

**Abstract**

**1. Introduction**

Bilge Doran, Hasan Orhun Köksal, Selen Aktan,

Bilge Doran, Hasan Orhun Köksal, Selen Aktan,

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **Constitutive Modeling of Masonry Walls Strengthened with Fiber Reinforced Polymers Constitutive Modeling of Masonry Walls Strengthened with Fiber Reinforced Polymers**

DOI: 10.5772/intechopen.77201

Bilge Doran, Hasan Orhun Köksal, Selen Aktan, Oktay Jafarov and Cengiz Karakoç Bilge Doran, Hasan Orhun Köksal, Selen Aktan, Oktay Jafarov and Cengiz Karakoç

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.77201

#### **Abstract**

This study primarily focuses on the constitutive modeling of masonry walls strengthened with fiber-reinforced polymers (FRP). In general, providing suitable relations for the material characterization of the masonry constituents so that the nonlinear finite element (NLFEA) applications of the elasto-plastic theory achieves a close fit to the experimental load-displacement diagrams of the walls subjected to in-plane loadings is desired. In this chapter, two relations proposed for masonry columns confined with FRP are adjusted for the cohesion and the internal friction angle of both units and mortar. Relating the mechanical parameters to the uniaxial compression strength and the hydrostatic pressure acting over the wall surface, the effects of major and intermediate principal stresses σ1 and σ<sup>2</sup> on the yielding and the shape of the deviatoric section are then reflected into the analyses. Finally, NLFEA results for the walls tested in different studies showed a good match with the experimental ones.

**Keywords:** masonry wall, fiber reinforced polymer, cohesion, internal friction, elasto-plastic analysis, finite element analysis

#### **1. Introduction**

Many historical masonry buildings of the world's architectural heritage are in dire need of retrofitting as they do not perform well to the seismic activities. Retrofitting of such structures needed better understanding of the constitutive features to characterize its present condition and determine the true structural safety for seismic actions. Although masonry has been used in building for centuries in many countries such as Turkey, constitutive

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

models and calculation techniques have not yet been available that allow realistic descriptions of the mechanic behavior of structures made of this heterogeneous material. During the last decade, the interest in seismic retrofitting of masonry structures has led to the development of specific engineering techniques and strategies in seismic areas. In this context, fiber-reinforced polymer (FRP) offered a capable alternative solution for retrofitting of masonry structures instead of conventional retrofitting techniques such as ferrocement, grout injection, and so on. Unreinforced masonry (URM) walls are the main load-bearing elements of masonry structures and the use of FRP enhances the strength, flexibility, and ductility under earthquake loading. Therefore, techniques of FRP application has become very effective in the strengthening of damaged URM walls. However, the strengthening of masonry structures by means of carbon fibers still lacks adequate theoretical and experimental verification, unlike reinforced concrete (RC) structures. Therefore, attention should be primarily focused on the description of the URM- and FRP-confined wall behavior under shear and compression. This study aims to investigate the finite element analysis (FEA) of URM walls strengthened with FRP composites by adopting an elasto-plastic approach proposed for the assessment of FRP-confined masonry columns under uniaxial compression [1].

The implementation of the plasticity theory to FRP-strengthened masonry walls under vertical and lateral loads by considering the constitutive behavior of masonry units, mortar, and FRP composite material separately, in this study, is analyzed. Köksal et al. [1] considered FRP-confined masonry columns as pressure-dependent material structures based on Drucker-Prager (DP) criterion. A DP-type yield criterion is also employed relating the cohesion and the internal friction angle of both the masonry unit and mortar to their uniaxial compressive strength by extending the previous approach to the FRP-strengthened masonry walls. For this purpose, an extensive parametric study is performed in order to explain the mechanical properties of masonry units and mortar with only one single parameter, that is, their uniaxial compressive strength [24]. Seven masonry walls are considered here. Three of them [25, 26] were unreinforced, single-sided strengthened with glass fiber-reinforced polymer (GFRP), and strengthened by carbon fiber-reinforced polymers (CFRP) strips on one side, respectively. Two of them [27] were strengthened by polyethylene terephthalate (PET) fiber sheets with unidirectional fiber; one was fully wrapped and the other was partially wrapped with both vertical and horizontal strips of PET sheets. One of them [28] was double-sided strengthened with FRP and the other [29] was unreinforced masonry walls, respectively. These seven walls are successively analyzed using FEA method with the proposed approach in LUSAS software [14].

Constitutive Modeling of Masonry Walls Strengthened with Fiber Reinforced Polymers

http://dx.doi.org/10.5772/intechopen.77201

5

A large literature has grown up which extends analytical and numerical models applied to masonry structures. These models involve mechanical parameters that are difficult to measure and cannot be implemented practically into numerical analyses. Several experimental and analytical studies on the behavior of masonry shear walls as the main structural component of masonry have been carried out in order to obtain a comprehensive approach for the overall assessment of masonry behavior [18, 21, 30–35]. Proposed theoretical models for implementation to FE analysis of masonry generally require a large number of material parameters that are difficult to measure easily and reliably. For this reason, several attempts have been made to express the stress-strain relationships of the masonry and its constituents using different modeling techniques such as micro-modeling, homogenization approach, and macro-modeling. The common approach in these studies is to treat masonry as a continuous medium or an equivalent continuous medium with the exception of detailed micro-modeling. Detailed micro-modeling should describe the masonry, mortar, and the interaction behavior between them [22, 36–38]. Köksal et al. [1, 8, 9] have performed FE analysis of masonry prisms and columns strengthened with FRP describing the constitutive behavior of the unit, mortar, and FRP separately. The similar approach has been adopted for the FE modeling of masonry shear walls while making the assumption that the material description of the mortar includes the effect of the unit-mortar interface behavior. Since the most common test performed on concrete-like brittle materials is for the measurement of its uniaxial compressive strength, it seems reasonable that the mechanical properties of masonry constituents can be assumed to be related to their compressive strengths. The authors have adopted this strategy in the elasto-plastic analyses of RC elements, masonry prisms, and columns, successively [1, 39, 40].

**2. The mechanics of constitutive modeling**

A large literature has grown up which extends analytical and numerical models applied to masonry structures. These models involve mechanical parameters that are difficult to measure and cannot be implemented practically into numerical analyses [2–13]. Researchers related to detailed micro-, simplified micro- and macro-FE modeling of masonry walls subjected to compression and shear have initially taken similar steps in the modeling process for masonry prisms and columns under concentric loading. Proceeding in a similar way with that used for the RC members [8, 9, 11, 12] employed a general-purpose finite element program LUSAS [14] for modeling the unreinforced and reinforced masonry prisms and columns activating both the elasto-plastic and isotropic damage models to reflect the nonlinear behavior of the blocks, grout, and mortar joints. Masonry walls were assumed as isotropic and elastic ignoring the influence of mortar joints acting as planes of weakness. The fact that a complete constitutive model should reflect the inelastic material properties of the units and mortar and a failure criterion describing the conditions for failure under combined compression-tension required more detailed analytical studies [15] and experimental investigations [16]. A micromodel should include several material parameters such as modulus of elasticity, Poisson's ratio, cohesion, internal friction angle, and compressive and tensile strength for the representation of units and mortar. Moreover, the composite unit-mortar interface model requires additional material parameters for the description of the interface behavior, for example, tensile strength, cohesion, internal friction angle, compressive strength, tensile fracture energy, shear fracture energy, dilatancy angle, and the close prediction for the experimental loaddisplacement diagram could be possible in the application of the theoretical model if the shear properties were reduced to 30%, the compressive strength to 20%, and the compressive fracture energy was multiplied by three [17]. From the engineering point of view, there is considerable limitation in the applicability of these sophisticated models [18–23] to FE analysis of masonry walls unless rational relations for the material parameters necessary to explain the models are provided.

The implementation of the plasticity theory to FRP-strengthened masonry walls under vertical and lateral loads by considering the constitutive behavior of masonry units, mortar, and FRP composite material separately, in this study, is analyzed. Köksal et al. [1] considered FRP-confined masonry columns as pressure-dependent material structures based on Drucker-Prager (DP) criterion. A DP-type yield criterion is also employed relating the cohesion and the internal friction angle of both the masonry unit and mortar to their uniaxial compressive strength by extending the previous approach to the FRP-strengthened masonry walls. For this purpose, an extensive parametric study is performed in order to explain the mechanical properties of masonry units and mortar with only one single parameter, that is, their uniaxial compressive strength [24]. Seven masonry walls are considered here. Three of them [25, 26] were unreinforced, single-sided strengthened with glass fiber-reinforced polymer (GFRP), and strengthened by carbon fiber-reinforced polymers (CFRP) strips on one side, respectively. Two of them [27] were strengthened by polyethylene terephthalate (PET) fiber sheets with unidirectional fiber; one was fully wrapped and the other was partially wrapped with both vertical and horizontal strips of PET sheets. One of them [28] was double-sided strengthened with FRP and the other [29] was unreinforced masonry walls, respectively. These seven walls are successively analyzed using FEA method with the proposed approach in LUSAS software [14].

### **2. The mechanics of constitutive modeling**

models and calculation techniques have not yet been available that allow realistic descriptions of the mechanic behavior of structures made of this heterogeneous material. During the last decade, the interest in seismic retrofitting of masonry structures has led to the development of specific engineering techniques and strategies in seismic areas. In this context, fiber-reinforced polymer (FRP) offered a capable alternative solution for retrofitting of masonry structures instead of conventional retrofitting techniques such as ferrocement, grout injection, and so on. Unreinforced masonry (URM) walls are the main load-bearing elements of masonry structures and the use of FRP enhances the strength, flexibility, and ductility under earthquake loading. Therefore, techniques of FRP application has become very effective in the strengthening of damaged URM walls. However, the strengthening of masonry structures by means of carbon fibers still lacks adequate theoretical and experimental verification, unlike reinforced concrete (RC) structures. Therefore, attention should be primarily focused on the description of the URM- and FRP-confined wall behavior under shear and compression. This study aims to investigate the finite element analysis (FEA) of URM walls strengthened with FRP composites by adopting an elasto-plastic approach proposed for the assessment of FRP-confined masonry columns under uniaxial

A large literature has grown up which extends analytical and numerical models applied to masonry structures. These models involve mechanical parameters that are difficult to measure and cannot be implemented practically into numerical analyses [2–13]. Researchers related to detailed micro-, simplified micro- and macro-FE modeling of masonry walls subjected to compression and shear have initially taken similar steps in the modeling process for masonry prisms and columns under concentric loading. Proceeding in a similar way with that used for the RC members [8, 9, 11, 12] employed a general-purpose finite element program LUSAS [14] for modeling the unreinforced and reinforced masonry prisms and columns activating both the elasto-plastic and isotropic damage models to reflect the nonlinear behavior of the blocks, grout, and mortar joints. Masonry walls were assumed as isotropic and elastic ignoring the influence of mortar joints acting as planes of weakness. The fact that a complete constitutive model should reflect the inelastic material properties of the units and mortar and a failure criterion describing the conditions for failure under combined compression-tension required more detailed analytical studies [15] and experimental investigations [16]. A micromodel should include several material parameters such as modulus of elasticity, Poisson's ratio, cohesion, internal friction angle, and compressive and tensile strength for the representation of units and mortar. Moreover, the composite unit-mortar interface model requires additional material parameters for the description of the interface behavior, for example, tensile strength, cohesion, internal friction angle, compressive strength, tensile fracture energy, shear fracture energy, dilatancy angle, and the close prediction for the experimental loaddisplacement diagram could be possible in the application of the theoretical model if the shear properties were reduced to 30%, the compressive strength to 20%, and the compressive fracture energy was multiplied by three [17]. From the engineering point of view, there is considerable limitation in the applicability of these sophisticated models [18–23] to FE analysis of masonry walls unless rational relations for the material parameters necessary to explain the

compression [1].

4 New Trends in Structural Engineering

models are provided.

A large literature has grown up which extends analytical and numerical models applied to masonry structures. These models involve mechanical parameters that are difficult to measure and cannot be implemented practically into numerical analyses. Several experimental and analytical studies on the behavior of masonry shear walls as the main structural component of masonry have been carried out in order to obtain a comprehensive approach for the overall assessment of masonry behavior [18, 21, 30–35]. Proposed theoretical models for implementation to FE analysis of masonry generally require a large number of material parameters that are difficult to measure easily and reliably. For this reason, several attempts have been made to express the stress-strain relationships of the masonry and its constituents using different modeling techniques such as micro-modeling, homogenization approach, and macro-modeling. The common approach in these studies is to treat masonry as a continuous medium or an equivalent continuous medium with the exception of detailed micro-modeling. Detailed micro-modeling should describe the masonry, mortar, and the interaction behavior between them [22, 36–38]. Köksal et al. [1, 8, 9] have performed FE analysis of masonry prisms and columns strengthened with FRP describing the constitutive behavior of the unit, mortar, and FRP separately. The similar approach has been adopted for the FE modeling of masonry shear walls while making the assumption that the material description of the mortar includes the effect of the unit-mortar interface behavior. Since the most common test performed on concrete-like brittle materials is for the measurement of its uniaxial compressive strength, it seems reasonable that the mechanical properties of masonry constituents can be assumed to be related to their compressive strengths. The authors have adopted this strategy in the elasto-plastic analyses of RC elements, masonry prisms, and columns, successively [1, 39, 40].

The most important mechanical parameter in FE analysis for masonry is the modulus of elasticity (*Em*) determined by uniaxial compression tests. The failure mechanism and load-displacement behavior of masonry are strongly affected by the differences in the modulus of elasticity between unit and mortar [24, 41]. In detailed micro-modeling, modulus of elasticity of units and mortar are required for NLFEA of the masonry walls. The relations for the masonry constituents are given in **Table 1** [24, 42].

Multi-axial stress states generally define the behavior of the structures such as masonry walls, RC panels, confined columns, or elements loaded over a specified limited area. The basic plastic models, that is, Mohr-Coulomb (MC) and DP are widely adopted in constitutive modeling of frictional materials, like concretes, soils, and rocks, when describing the material behavior beyond the elastic range. Since material parameters of MC criterion are easily obtained from standard tests and are logical from the physical point of view, MC criterion is the most common used criterion in geotechnical and structural engineering [13]. MC criterion in threedimensional (3D) stress space can be expressed as Eq. (1):

$$\begin{aligned} \text{dimensional (3D) stress space can be expressed as Eq. (1):}\\ \text{\(\xi, \rho, \theta\) = \sqrt{2} \ \xi \sin \phi + \sqrt{3} \ \rho \sin \left(\theta + \frac{\pi}{3}\right) + \rho \cos \left(\theta + \frac{\pi}{3}\right) \sin \phi - \sqrt{6} \ c \cos \phi &= 0 \\ \text{for } 0 \prec \theta \prec \frac{\pi}{3} \end{aligned} \tag{1}$$

where *ξ* = *I* 1 /√ \_\_ 3 *ξ* = *I* 1 /√ \_\_ 3and *ρ* = √ \_\_ 2 *J* 2 [43]. The invariants *ξ* and *I*<sup>1</sup> indicate the hydrostatic component of the current stress state. However, because of the smooth surface approximation to the six-faceted MC yield function, DP criterion is conversely easy to implement and allows a fast computation of plastic behavior, even it is a drawback from the physical point of view given as Eq. (2) [1, 44]:

$$f(\xi, \rho) = \sqrt{6} \, a\xi + \rho - \sqrt{2} \, k = 0 \tag{2}$$

In masonry columns strengthened with FRP, since the existence of major and intermediate

effect of FRP confinement should be included in the derivation of the material parameters. Due to the complexity of establishing a general failure surface in the principal stress space

The cohesion and the internal friction angle of the units to include the confinement stresses

\_\_\_\_ |*σ* \_\_\_\_*m*| *f mu*

\_\_\_\_*m*| *f mu*

For high hydrostatic pressures (0 > *σm/fmu ≥ −1)*, internal friction angle can be taken as a constant value, that is, 30°–35° for concrete and nearly 45° for clay brick as illustrated in **Figure 1**. Since much lower pressures exist on masonry walls and their constituents, material parameters have been calibrated to best reconcile the experimental data of several walls. In the wall tests, as the axial load level is less than the ultimate uniaxial strength, the mean stress can be taken as a

finally verified by numerical simulations of the walls subjected to the pressures in the range of

The use of the compressive meridian obviously causes some overestimation for the failure load for this case. In order to lessen this overestimation, an ending point definition for the termination of NLFEA should be needed. The ending point for the analysis can be determined

≤ *c* \_\_\_*mu f mu* ||*σ* \_\_\_\_*m*| *f mu*

<sup>≥</sup> *<sup>φ</sup><sup>f</sup>* <sup>=</sup> *<sup>φ</sup>mu*||*<sup>I</sup>*

can be nearly equal to *π/3* for brick and *π/4* for concrete. Besides, applying the

\_\_\_1| *f mu*

is approximately *π/4* for brick and between *π/5* and *π/6* for concrete.

)/ 3 at all stress levels. The validity of Eqs. (4) and (5) is

because of the lack of necessary experimental data, the mean stress *σ<sup>m</sup>* = *I*

<sup>=</sup> tan*<sup>φ</sup>* \_\_\_\_\_\_*mu* 3 √

*l*

point is defined as a constant value which is corresponding to (*f*

\_\_\_*mu f mu*

will affect the yielding and the shape of the deviatoric section, the

Constitutive Modeling of Masonry Walls Strengthened with Fiber Reinforced Polymers

1

http://dx.doi.org/10.5772/intechopen.77201

= 1 (4)

= 1 (5)

*mu* + 2*f l* /3 at the failure

7

)/3 at all stress levels.

principal stresses *σ*<sup>1</sup>

are given as Eq. (4):

In Eq. (5), *φ<sup>i</sup>*

*ξ*/*f*

boundary condition, *φ<sup>f</sup>*

*<sup>c</sup>*

and *σ*<sup>2</sup>

*<sup>φ</sup>mu* <sup>=</sup> *<sup>φ</sup><sup>i</sup>* <sup>−</sup> 0.75 <sup>|</sup>*<sup>σ</sup>*

constant value corresponding to (*p* + 2*f*

*mu* ≤ −0.58 throughout this study.

**Figure 1.** Plotting of Eq. (5) for the internal friction angle of clay brick [13].

where *α* and *k* are material constants. Generally, these constants are expressed in terms of the cohesion (*c*) and the internal friction angle (*φ*) [13]. Therefore, the surfaces of both DP and MC yield criteria are made to coincide along the compression meridian, and then the constants *α* and *k* are related to the constants *c* and *φ* by [45] given in Eq. (3):

$$\alpha = \frac{2\sin\phi}{\sqrt{3}(3-\sin\phi)}, k = \frac{6c\cos\phi}{\sqrt{3}(3-\sin\phi)}\tag{3}$$


**Table 1.** Expressions for modulus of elasticity of the masonry constituents.

In masonry columns strengthened with FRP, since the existence of major and intermediate principal stresses *σ*<sup>1</sup> and *σ*<sup>2</sup> will affect the yielding and the shape of the deviatoric section, the effect of FRP confinement should be included in the derivation of the material parameters. Due to the complexity of establishing a general failure surface in the principal stress space because of the lack of necessary experimental data, the mean stress *σ<sup>m</sup>* = *I* 1 /3 at the failure point is defined as a constant value which is corresponding to (*f mu* + 2*f l* )/3 at all stress levels. The cohesion and the internal friction angle of the units to include the confinement stresses are given as Eq. (4):

The most important mechanical parameter in FE analysis for masonry is the modulus of elasticity (*Em*) determined by uniaxial compression tests. The failure mechanism and load-displacement behavior of masonry are strongly affected by the differences in the modulus of elasticity between unit and mortar [24, 41]. In detailed micro-modeling, modulus of elasticity of units and mortar are required for NLFEA of the masonry walls. The relations for the masonry constituents are

Multi-axial stress states generally define the behavior of the structures such as masonry walls, RC panels, confined columns, or elements loaded over a specified limited area. The basic plastic models, that is, Mohr-Coulomb (MC) and DP are widely adopted in constitutive modeling of frictional materials, like concretes, soils, and rocks, when describing the material behavior beyond the elastic range. Since material parameters of MC criterion are easily obtained from standard tests and are logical from the physical point of view, MC criterion is the most common used criterion in geotechnical and structural engineering [13]. MC criterion in three-

given in **Table 1** [24, 42].

6 New Trends in Structural Engineering

*<sup>f</sup>*(*ξ*, *<sup>ρ</sup>*, *<sup>θ</sup>*) <sup>=</sup> <sup>√</sup>

where *ξ* = *I*

3

1 /√ \_\_ 3 *ξ* = *I* 1 /√ \_\_

given as Eq. (2) [1, 44]:

dimensional (3D) stress space can be expressed as Eq. (1):

\_\_

\_\_ 2 *J* 2

and *k* are related to the constants *c* and *φ* by [45] given in Eq. (3):

√ \_\_ 3(3 − sin*φ*)

<sup>3</sup> *<sup>ρ</sup>* sin(*<sup>θ</sup>* <sup>+</sup> \_\_

*π*

*for* <sup>0</sup> <sup>≺</sup> *<sup>θ</sup>* <sup>≺</sup> \_\_

[43]. The invariants *ξ* and *I*<sup>1</sup>

6 + *ρ* − √

where *α* and *k* are material constants. Generally, these constants are expressed in terms of the cohesion (*c*) and the internal friction angle (*φ*) [13]. Therefore, the surfaces of both DP and MC yield criteria are made to coincide along the compression meridian, and then the constants *α*

\_\_

, *<sup>k</sup>* <sup>=</sup> <sup>6</sup>*<sup>c</sup>* cos*<sup>φ</sup>* \_\_\_\_\_\_\_\_ √ \_\_

*bl*

*mr*

*br*

ponent of the current stress state. However, because of the smooth surface approximation to the six-faceted MC yield function, DP criterion is conversely easy to implement and allows a fast computation of plastic behavior, even it is a drawback from the physical point of view

\_\_

<sup>3</sup> ) <sup>+</sup> *<sup>ρ</sup>* cos(*<sup>θ</sup>* <sup>+</sup> \_\_

*π*

<sup>3</sup> ) sin*φ* − √

\_\_

6 *c* cos*φ* = 0

indicate the hydrostatic com-

2 *k* = 0 (2)

<sup>3</sup>(<sup>3</sup> <sup>−</sup> sin*φ*) (3)

(1)

2 *ξ* sin*φ* + √

3and *ρ* = √

\_\_

*π*

*f*(*ξ*, *ρ*) = √

*<sup>α</sup>* <sup>=</sup> <sup>2</sup> sin*<sup>φ</sup>* \_\_\_\_\_\_\_\_

**Material Expression** Brick *Ebr* = (300~700) *f*

Block *Ebl* = 1000 *f*

Mortar *Emr* = 200 *f*

**Table 1.** Expressions for modulus of elasticity of the masonry constituents.

$$\frac{c\_{nu}}{f\_{mu}} = \frac{\tan \phi\_{nu}}{3} \sqrt{\frac{|\sigma\_n|}{f\_{nu}}} \le \frac{c\_{nu}}{f\_{nu}} \Big| \frac{|\sigma\_n|}{f\_{nu}} = \mathbf{1} \tag{4}$$

$$
\phi\_{mu} = \phi\_l - 0.75 \frac{|\sigma\_n|}{f\_{mu}} \ge \phi\_f = \phi\_{mu} \left| \frac{|I\_i|}{f\_{mu}} = 1 \right. \tag{5}
$$

In Eq. (5), *φ<sup>i</sup>* can be nearly equal to *π/3* for brick and *π/4* for concrete. Besides, applying the boundary condition, *φ<sup>f</sup>* is approximately *π/4* for brick and between *π/5* and *π/6* for concrete. For high hydrostatic pressures (0 > *σm/fmu ≥ −1)*, internal friction angle can be taken as a constant value, that is, 30°–35° for concrete and nearly 45° for clay brick as illustrated in **Figure 1**. Since much lower pressures exist on masonry walls and their constituents, material parameters have been calibrated to best reconcile the experimental data of several walls. In the wall tests, as the axial load level is less than the ultimate uniaxial strength, the mean stress can be taken as a constant value corresponding to (*p* + 2*f l* )/ 3 at all stress levels. The validity of Eqs. (4) and (5) is finally verified by numerical simulations of the walls subjected to the pressures in the range of *ξ*/*f mu* ≤ −0.58 throughout this study.

The use of the compressive meridian obviously causes some overestimation for the failure load for this case. In order to lessen this overestimation, an ending point definition for the termination of NLFEA should be needed. The ending point for the analysis can be determined

**Figure 1.** Plotting of Eq. (5) for the internal friction angle of clay brick [13].

**Figure 2.** Assumption for the discontinuity surface between two units along the thickness of the mortar resulting failure of the wall [13].

defining the discontinuity surface between two units along the thickness of the mortar as illustrated in **Figure 2**. The maximum tensile strain can then be determined from the case of that a complete opening will occur along the mortar thickness between two units as given in Eq. (6):

$$
\varepsilon\_{\text{max}} = \left| h\_{nr} / \left( 2 \, h\_{br} + h\_{nr} \right) \right. \tag{6}
$$

**Table 2** shows the material and geometrical properties for masonry walls with its FRP confinement evaluated in this study. All mechanical properties of the masonry constituents uti-

**Figure 3.** (a) Eight-noded hexahedral element for units and mortar and (b) four-noded thick shell element for FRP jacket [13].

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Stratford et al. [25] tested six 1200 × 1200 mm masonry panels under a combination of vertical preloading, and in-plane horizontal shear loading. Both clay and concrete brick specimens were tested while one specimen of each material was left unreinforced and the other two panels were single-sided strengthened using GFRP. GFRP had equal amounts of fibers in the horizontal and vertical directions (parallel to the mortar joints) and fibers were oriented at 45° to the joints. As can be seen in **Figure 4**, two connected hydraulic jacks (N) were placed on the

For the case of URM wall, the end point for the FEA is determined defining a discontinuity surface between two clay units along the thickness of the mortar. The maximum tensile strain pointing out that a complete opening will occur along the mortar thickness between two units, can be found by dividing the mortar joint thickness 10 mm by two block heights plus the mortar thickness as 10/(2 × 65 + 10) = 0.071. This strain value can be called critical tensile strain. As the lateral load is increased, the effect of tensile deformations grows up along the toe of the wall and it is easy to observe the critical tensile strain at the toe of the wall from **Figure 5**. If FEA ends when the ultimate tensile strain at the wall reaches the critical value, both simulation model and laboratory test results agreed very well as shown in **Figure 6**.

As can be seen in **Figure 6**, horizontal displacement at the failure is experimentally determined approximately as 14 mm. NLFEA gives a value of 13.81 mm from the use of the maximum tensile strain criterion for the wall. As can be seen in **Figures 5** and **7**, the region of concentrated maximum tensile strain matches the location of experimental crack pattern on the wall.

**Figure 8** shows the stress-strain plots for strengthened wall Clay 2. The load-displacement plot obtained from NLFEA is well agreed with the experimental data. As shown in **Figure 9**, the analysis is terminated upon reaching the maximum tensile strain corresponding to the tensile strength of GFRP at a horizontal drift of 10.16 mm. This value is experimentally determined

wall while the shear load was applied to the wall by a horizontal hydraulic jack (P).

lized in the analyses are also given in **Table 3**.

approximately as 12 mm.

Mortar generally governs the nonlinear behavior of masonry and has an influence on the axial strain of masonry prism and also on the shear response of the masonry walls [24, 46]. Therefore, any theoretical model should account for the nonlinear response of both the mortar and the unit-mortar interface to predict the inelastic behavior of masonry [41, 47]. In this study, the mechanical parameters of mortar, given in Eqs. (7) and (8), are adjusted to reflect the unit-mortar interface response as preferred in the modeling of masonry prisms and columns previously [1, 8]:

$$c\_{\rm uv} = 1.55 \sqrt[3]{f\_{\rm uv}} \tag{7}$$

$$
\phi\_{\rm uv} = 1.519 f\_{\rm uv} \tag{8}
$$

#### **3. Finite element modeling and model verification**

Three-dimensional (3D) finite element models for masonry walls strengthened with FRP composites are developed in LUSAS [14]. Masonry constituents, that is, brick and mortar, are assumed to be isotropic elasto-plastic obeying DP criterion. They are modeled separately with eight-noded hexahedral element (HX8M) which is a solid element with an incompatible strain field as illustrated in **Figure 3a**. FRP composites are modeled by four-noded thick shell element (QTS4) which has a thick and thin curved shell geometry including multiple branched junctions as in **Figure 3b**. Both the element formulations take account of membrane, shear and flexural deformations, and are capable of modeling inelastic phenomenon.

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**Figure 3.** (a) Eight-noded hexahedral element for units and mortar and (b) four-noded thick shell element for FRP jacket [13].

defining the discontinuity surface between two units along the thickness of the mortar as illustrated in **Figure 2**. The maximum tensile strain can then be determined from the case of that a complete opening will occur along the mortar thickness between two units as given in

**Figure 2.** Assumption for the discontinuity surface between two units along the thickness of the mortar resulting failure

*ε*max = *hmr*/(2 *hbr* + *hmr*) (6)

Mortar generally governs the nonlinear behavior of masonry and has an influence on the axial strain of masonry prism and also on the shear response of the masonry walls [24, 46]. Therefore, any theoretical model should account for the nonlinear response of both the mortar and the unit-mortar interface to predict the inelastic behavior of masonry [41, 47]. In this study, the mechanical parameters of mortar, given in Eqs. (7) and (8), are adjusted to reflect the unit-mortar interface response as preferred in the modeling of masonry prisms and col-

> √ \_\_\_ *f*

Three-dimensional (3D) finite element models for masonry walls strengthened with FRP composites are developed in LUSAS [14]. Masonry constituents, that is, brick and mortar, are assumed to be isotropic elasto-plastic obeying DP criterion. They are modeled separately with eight-noded hexahedral element (HX8M) which is a solid element with an incompatible strain field as illustrated in **Figure 3a**. FRP composites are modeled by four-noded thick shell element (QTS4) which has a thick and thin curved shell geometry including multiple branched junctions as in **Figure 3b**. Both the element formulations take account of membrane, shear and

*mr* (7)

*mr* (8)

Eq. (6):

of the wall [13].

8 New Trends in Structural Engineering

umns previously [1, 8]:

*cmr* <sup>=</sup> 1.55 <sup>3</sup>

*φmr* = 1.519 *f*

**3. Finite element modeling and model verification**

flexural deformations, and are capable of modeling inelastic phenomenon.

**Table 2** shows the material and geometrical properties for masonry walls with its FRP confinement evaluated in this study. All mechanical properties of the masonry constituents utilized in the analyses are also given in **Table 3**.

Stratford et al. [25] tested six 1200 × 1200 mm masonry panels under a combination of vertical preloading, and in-plane horizontal shear loading. Both clay and concrete brick specimens were tested while one specimen of each material was left unreinforced and the other two panels were single-sided strengthened using GFRP. GFRP had equal amounts of fibers in the horizontal and vertical directions (parallel to the mortar joints) and fibers were oriented at 45° to the joints. As can be seen in **Figure 4**, two connected hydraulic jacks (N) were placed on the wall while the shear load was applied to the wall by a horizontal hydraulic jack (P).

For the case of URM wall, the end point for the FEA is determined defining a discontinuity surface between two clay units along the thickness of the mortar. The maximum tensile strain pointing out that a complete opening will occur along the mortar thickness between two units, can be found by dividing the mortar joint thickness 10 mm by two block heights plus the mortar thickness as 10/(2 × 65 + 10) = 0.071. This strain value can be called critical tensile strain. As the lateral load is increased, the effect of tensile deformations grows up along the toe of the wall and it is easy to observe the critical tensile strain at the toe of the wall from **Figure 5**. If FEA ends when the ultimate tensile strain at the wall reaches the critical value, both simulation model and laboratory test results agreed very well as shown in **Figure 6**.

As can be seen in **Figure 6**, horizontal displacement at the failure is experimentally determined approximately as 14 mm. NLFEA gives a value of 13.81 mm from the use of the maximum tensile strain criterion for the wall. As can be seen in **Figures 5** and **7**, the region of concentrated maximum tensile strain matches the location of experimental crack pattern on the wall.

**Figure 8** shows the stress-strain plots for strengthened wall Clay 2. The load-displacement plot obtained from NLFEA is well agreed with the experimental data. As shown in **Figure 9**, the analysis is terminated upon reaching the maximum tensile strain corresponding to the tensile strength of GFRP at a horizontal drift of 10.16 mm. This value is experimentally determined approximately as 12 mm.


**Table 2.** General characteristics of masonry walls and FRP confinement.


**Table 3.** Mechanical properties adopted for FE analyses.

In all experiments, except for the unreinforced clay specimen (Clay 1), the strengthened masonry walls (Clay 2) failed by rapid propagation of diagonal crack which followed the mortar joints as in **Figure 10a**. Crushing failure, which occurred when the compressive strength of the mortar was reached, cannot happen for these walls because of the mortar strength higher than 11 MPa. Shear failure, involving sliding along a slip plane either within the mortar or at the brick-mortar interface, is the main failure mode. The fracture pattern is well predicted through the concentration of the maximum tensile strains diagonally shown in **Figure 10b**.

strengthened wall was subjected to cyclic loading until failure. Specimens were built with clay bricks and mortar with thickness 4 mm. Flexural tensile strength of mortar was 0.80 MPa. A single-story HRM C2 wall was built using historic full clay bricks in scale, one-third, and was tested in a special frame (**Figure 11**). Besides, mechanical and geometrical properties of brick,

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Initially, pre-compression load (1.50 MPa) was applied to the flanges and web by three jacks, kept constant, then horizontal force was applied measuring horizontal force until failure [26]. Wall C2 damaged by cyclic shear tests and then was strengthened by 50 mm wide carbon strips

mortar and CFRP strips are given here in **Tables 2** and **3**.

**Figure 5.** Maximum strains along the wall height for Clay 1 [13].

**Figure 4.** Experimental setup of shear wall Clay 2 [13].

Capozucca [26] investigated the behavior of historical reinforced masonry (HRM) walls strengthened by CFRP strips, experimentally. CFRP strips bonded to only one face of the specimen horizontally and vertically. FRP strips, containing unidirectional fibers, can be bonded to the surface of the wall and arranged to provide an external truss action [25]. The Constitutive Modeling of Masonry Walls Strengthened with Fiber Reinforced Polymers http://dx.doi.org/10.5772/intechopen.77201 11

**Figure 4.** Experimental setup of shear wall Clay 2 [13].

**Figure 5.** Maximum strains along the wall height for Clay 1 [13].

In all experiments, except for the unreinforced clay specimen (Clay 1), the strengthened masonry walls (Clay 2) failed by rapid propagation of diagonal crack which followed the mortar joints as in **Figure 10a**. Crushing failure, which occurred when the compressive strength of the mortar was reached, cannot happen for these walls because of the mortar strength higher than 11 MPa. Shear failure, involving sliding along a slip plane either within the mortar or at the brick-mortar interface, is the main failure mode. The fracture pattern is well predicted through the concentration of the maximum tensile strains diagonally shown

*<sup>E</sup>mu* **(MPa)** *<sup>c</sup>mu* **(MPa)** *φmu* **(°)** *<sup>E</sup>mr* **(MPa)** *<sup>c</sup>mr* **(MPa)** *φmr* **(°)**

Clay 2 GFRP 18,600 3.04 59.68 3250 3.45 16.71

**Wall dimension** 

Clay 2 GFRP 1.38 1200 × 1200 × 60 228 × 65 × 60 73,300 0.15

WALL 3 1.16 860 × 540 × 100 205 × 60 × 100 10,000 0.841

W4-C-RT 1.29 1830 × 1830 × 203 406 × 152 × 203 96,500 1.60

**Brick dimension (mm)**

**FRP** *E***FRP (MPa)**

*t* **(mm)**

**(mm)**

Stratford [25] Clay 1 URM 1.38 1200 × 1200 × 60 228 × 65 × 60 - -

Haider [29] URM 6 0.5 2870 × 2408 × 150 310 × 76 × 150 - -

Capozucca [26] HRM-C2 1.5 840 × 633 × 50 100 × 17 × 50 240,000 0.177

WALL 4 8000 2.16 59.06 125 12.5 18.90

**Reference Experiment Brick Mortar**

**Table 2.** General characteristics of masonry walls and FRP confinement.

**load (MPa)**

Stratford [25] Clay1 URM 18,600 3.04 59.68 3250 3.45 16.71

Capozucca [26] HRM-C2 7000 2.33 59.37 150 2.21 4.41 Rahman and Ueda [27] WALL 3 8000 2.27 58.94 125 12.5 18.90

Mosallam and Banerjee [28] W4-C-RT 6000 2.32 57.66 800 3.79 22.20 Haider [29] URM 6 20,000 0.90 59.54 1100 2.65 7.59

Capozucca [26] investigated the behavior of historical reinforced masonry (HRM) walls strengthened by CFRP strips, experimentally. CFRP strips bonded to only one face of the specimen horizontally and vertically. FRP strips, containing unidirectional fibers, can be bonded to the surface of the wall and arranged to provide an external truss action [25]. The

in **Figure 10b**.

**Table 3.** Mechanical properties adopted for FE analyses.

**Reference Experiment Precompression** 

10 New Trends in Structural Engineering

WALL 4

Rahman and Ueda [27]

Mosallam and Banerjee [28]

> strengthened wall was subjected to cyclic loading until failure. Specimens were built with clay bricks and mortar with thickness 4 mm. Flexural tensile strength of mortar was 0.80 MPa. A single-story HRM C2 wall was built using historic full clay bricks in scale, one-third, and was tested in a special frame (**Figure 11**). Besides, mechanical and geometrical properties of brick, mortar and CFRP strips are given here in **Tables 2** and **3**.

> Initially, pre-compression load (1.50 MPa) was applied to the flanges and web by three jacks, kept constant, then horizontal force was applied measuring horizontal force until failure [26]. Wall C2 damaged by cyclic shear tests and then was strengthened by 50 mm wide carbon strips

**Figure 6.** Lateral load-displacement responses for Clay 1 [13].

**Figure 7.** Experimental crack pattern for Clay 1 [13].

bonded to one side, parallel and normal to mortar joints. Mechanical parameters of CFRP strips are summarized in **Table 2**. Walls C1 and C2 were initially subjected, respectively, to five cyclic stages of horizontal force and seven cyclic stages before to bring the walls to failure by an ultimate stage with unidirectional horizontal force [26]. These are the possible reasons for the stiffer behavior observed in **Figures 12** and **13**. The critical tensile strain assumption for masonry is adopted for the stress-strain plots (**Figure 13**), while the analysis is terminated by reaching the tensile strength of GFRP as in **Figure 12**. Therefore, the material parameters for the masonry constituents of the pre-damaged walls should be somewhat lower than the recommended ones in Eqs. (4)–(7). The cohesion and internal friction angle values of the clay brick lowered by 35 and 15%, respectively. Since the mortar has already defined by a very weak material, any reduction on its mechanical properties will not be reasonable. Another reason for the stiffer predictions can be the present approach does not consider the debonding

**Figure 10.** (a) Schematic crack patterns in strengthened wall (Clay 2) and (b) maximum tensile strains over unreinforced

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**Figure 9.** Maximum strain distribution over GFRP surface [13].

face in 3D FE model [13].

**Figure 8.** Lateral load-displacement responses for strengthened Clay 2 [13].

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**Figure 9.** Maximum strain distribution over GFRP surface [13].

**Figure 6.** Lateral load-displacement responses for Clay 1 [13].

12 New Trends in Structural Engineering

**Figure 7.** Experimental crack pattern for Clay 1 [13].

**Figure 8.** Lateral load-displacement responses for strengthened Clay 2 [13].

**Figure 10.** (a) Schematic crack patterns in strengthened wall (Clay 2) and (b) maximum tensile strains over unreinforced face in 3D FE model [13].

bonded to one side, parallel and normal to mortar joints. Mechanical parameters of CFRP strips are summarized in **Table 2**. Walls C1 and C2 were initially subjected, respectively, to five cyclic stages of horizontal force and seven cyclic stages before to bring the walls to failure by an ultimate stage with unidirectional horizontal force [26]. These are the possible reasons for the stiffer behavior observed in **Figures 12** and **13**. The critical tensile strain assumption for masonry is adopted for the stress-strain plots (**Figure 13**), while the analysis is terminated by reaching the tensile strength of GFRP as in **Figure 12**. Therefore, the material parameters for the masonry constituents of the pre-damaged walls should be somewhat lower than the recommended ones in Eqs. (4)–(7). The cohesion and internal friction angle values of the clay brick lowered by 35 and 15%, respectively. Since the mortar has already defined by a very weak material, any reduction on its mechanical properties will not be reasonable. Another reason for the stiffer predictions can be the present approach does not consider the debonding

**Figure 11.** Geometrical details of HRM C2 wall [13, 26].

**Figure 13.** Comparison of load-displacement responses for pre-damaged HRM C2 wall adopting the maximum tensile

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**Figure 14.** Geometrical details of experiments (a) WALL 3 and (b) WALL 4 [27].

**Figure 15.** Comparison of load-displacement responses for WALL 3.

strain criterion of masonry [13].

**Figure 12.** Comparison of load-displacement responses for pre-damaged HRM C2 wall employing the ultimate tensile strain criterion of GFRP [13].

phenomenon between the wall and FRP layer which affects the maximum lateral displacement of the wall particularly while reaching the ultimate load capacity [48].

Actually, presence of discontinuity surfaces with some specific values of separation affects the failure mechanisms and the level of FRP confinement in cracked and damaged walls. Therefore, the use of the critical tensile strain criterion instead of the ultimate tensile strength of FRP can be more reasonable for the analysis of pre-damaged walls.

Rahman and Ueda [27] studied the mechanical behavior of masonry walls experimentally. WALLS 1 and 2 were unreinforced, and WALLS 3 and 4 were fully (%100) and partially (%40) strengthened with FRP, respectively (**Figure 14**). The dimensions of each brick unit were 205 × 100 × 60 mm while the dimensions of walls were 860 × 540 × 100mm with an approximate mortar thickness of 10 mm. A pre-compression of 100 kN (1.16 MPa) was Constitutive Modeling of Masonry Walls Strengthened with Fiber Reinforced Polymers http://dx.doi.org/10.5772/intechopen.77201 15

**Figure 13.** Comparison of load-displacement responses for pre-damaged HRM C2 wall adopting the maximum tensile strain criterion of masonry [13].

**Figure 14.** Geometrical details of experiments (a) WALL 3 and (b) WALL 4 [27].

phenomenon between the wall and FRP layer which affects the maximum lateral displace-

**Figure 12.** Comparison of load-displacement responses for pre-damaged HRM C2 wall employing the ultimate tensile

Actually, presence of discontinuity surfaces with some specific values of separation affects the failure mechanisms and the level of FRP confinement in cracked and damaged walls. Therefore, the use of the critical tensile strain criterion instead of the ultimate tensile strength

Rahman and Ueda [27] studied the mechanical behavior of masonry walls experimentally. WALLS 1 and 2 were unreinforced, and WALLS 3 and 4 were fully (%100) and partially (%40) strengthened with FRP, respectively (**Figure 14**). The dimensions of each brick unit were 205 × 100 × 60 mm while the dimensions of walls were 860 × 540 × 100mm with an approximate mortar thickness of 10 mm. A pre-compression of 100 kN (1.16 MPa) was

ment of the wall particularly while reaching the ultimate load capacity [48].

of FRP can be more reasonable for the analysis of pre-damaged walls.

**Figure 11.** Geometrical details of HRM C2 wall [13, 26].

14 New Trends in Structural Engineering

strain criterion of GFRP [13].

**Figure 15.** Comparison of load-displacement responses for WALL 3.

applied on the top of the wall. The modulus of elasticity and the tensile strength of FRP sheet were 10,000 and 740 MPa, respectively [49].

**Figure 16.** Comparison of load-displacement responses for WALL 4.

**Figure 17.** Maximum strains along the height of WALL 3.

The stress-strain plots for WALL 3 are shown in **Figure 15**. The load-displacement plot (**Figure 16**) obtained from NLFEA is well agreed with the experimental data. The analysis is terminated upon reaching the maximum tensile strain corresponding to the tensile strength of PET at a horizontal drift of 8.80 mm for WALL 3 (**Figure 17**). This value is experimentally determined approximately as 9.30 mm. Finally, the maximum strains along the wall height are depicted in **Figure 17**. The analysis is terminated upon reaching the maximum tensile strain corresponding to the tensile strength of PET at a horizontal drift of 12.50 mm (**Figure 18**). This value is experimentally determined approximately as 12.40 mm for WALL 4. In **Figure 18**, the maximum strains along the wall

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height can be seen.

**Figure 19.** Experimental setup of W4-C-RT wall.

**Figure 20.** Lateral load-displacement responses for W4-C-RT.

**Figure 18.** Maximum strains along the height of WALL 4.

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**Figure 19.** Experimental setup of W4-C-RT wall.

applied on the top of the wall. The modulus of elasticity and the tensile strength of FRP

sheet were 10,000 and 740 MPa, respectively [49].

16 New Trends in Structural Engineering

**Figure 16.** Comparison of load-displacement responses for WALL 4.

**Figure 17.** Maximum strains along the height of WALL 3.

**Figure 18.** Maximum strains along the height of WALL 4.

**Figure 20.** Lateral load-displacement responses for W4-C-RT.

The stress-strain plots for WALL 3 are shown in **Figure 15**. The load-displacement plot (**Figure 16**) obtained from NLFEA is well agreed with the experimental data. The analysis is terminated upon reaching the maximum tensile strain corresponding to the tensile strength of PET at a horizontal drift of 8.80 mm for WALL 3 (**Figure 17**). This value is experimentally determined approximately as 9.30 mm. Finally, the maximum strains along the wall height are depicted in **Figure 17**. The analysis is terminated upon reaching the maximum tensile strain corresponding to the tensile strength of PET at a horizontal drift of 12.50 mm (**Figure 18**). This value is experimentally determined approximately as 12.40 mm for WALL 4. In **Figure 18**, the maximum strains along the wall height can be seen.

**Figure 21.** Maximum strains along the wall height (W4-C-RT).

**Figure 22.** Experimental setup of URM 6 wall.

Mosallam and Banerjee [28] tested six identical masonry wall specimens of size 1830 × 1830 mm with 406 × 152 × 203 mm hollow concrete block under a combination of constant axial and incremental lateral loads (push-pull) (**Figure 19**). Two of them were unreinforced and the others were strengthened with different types of FRP elements. The compressive strength of masonry prisms, grout cylinders, and mortar cylinders was calculated as 2.16, 18.96, and 14.62 MPa, respectively. The tensile strength of FRP was 1061 MPa and modulus of elasticity was 96,500 MPa. In this study, W4-C-RT wall retrofitted wall specimen with doublesided carbon/epoxy laminate is considered. The rapture strain of FRP element is calculated as 0.011 from ratio of tensile stress of FRP to modulus of elasticity. Load-displacement curves obtained from experiments and numerical analysis are given in **Figure 20**.

The load-displacement plot obtained from NLFEA is well agreed with the experimental data. As shown in **Figure 21**, the analysis is terminated upon reaching the maximum tensile strain corresponding to the tensile strength of FRP at a horizontal drift of 8.20 mm. This value is experimentally determined approximately as 8.00 mm. The critical tensile strain is calculated as 0.038. As the lateral load is increased, the effect of tensile deformations grows up along the toe of the wall and it is easy to observe the critical tensile strain at the toe of the wall from

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Haider [29] investigated the behavior of clay brick masonry walls under a combination of vertical preloading, and lateral shear loading (**Figure 22**). Mechanical and geometrical properties of brick and mortar are given in **Table 3**. Compressive strength of clay brick and mortar were 15.7 and 5.00 MPa, respectively. For the case of clay brick masonry wall, the end point for the

**Figure 21**.

**Figure 23.** Lateral load-displacement responses for URM 6.

**Figure 24.** Maximum strains along the wall height (URM 6).

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**Figure 23.** Lateral load-displacement responses for URM 6.

**Figure 24.** Maximum strains along the wall height (URM 6).

Mosallam and Banerjee [28] tested six identical masonry wall specimens of size 1830 × 1830 mm with 406 × 152 × 203 mm hollow concrete block under a combination of constant axial and incremental lateral loads (push-pull) (**Figure 19**). Two of them were unreinforced and the others were strengthened with different types of FRP elements. The compressive strength of masonry prisms, grout cylinders, and mortar cylinders was calculated as 2.16, 18.96, and 14.62 MPa, respectively. The tensile strength of FRP was 1061 MPa and modulus of elasticity was 96,500 MPa. In this study, W4-C-RT wall retrofitted wall specimen with doublesided carbon/epoxy laminate is considered. The rapture strain of FRP element is calculated as 0.011 from ratio of tensile stress of FRP to modulus of elasticity. Load-displacement curves

obtained from experiments and numerical analysis are given in **Figure 20**.

**Figure 21.** Maximum strains along the wall height (W4-C-RT).

18 New Trends in Structural Engineering

**Figure 22.** Experimental setup of URM 6 wall.

The load-displacement plot obtained from NLFEA is well agreed with the experimental data. As shown in **Figure 21**, the analysis is terminated upon reaching the maximum tensile strain corresponding to the tensile strength of FRP at a horizontal drift of 8.20 mm. This value is experimentally determined approximately as 8.00 mm. The critical tensile strain is calculated as 0.038. As the lateral load is increased, the effect of tensile deformations grows up along the toe of the wall and it is easy to observe the critical tensile strain at the toe of the wall from **Figure 21**.

Haider [29] investigated the behavior of clay brick masonry walls under a combination of vertical preloading, and lateral shear loading (**Figure 22**). Mechanical and geometrical properties of brick and mortar are given in **Table 3**. Compressive strength of clay brick and mortar were 15.7 and 5.00 MPa, respectively. For the case of clay brick masonry wall, the end point for the FEA is determined defining a discontinuity surface between two brick units along the thickness of the mortar. The maximum tensile strain can be found as 0.061. Load-displacement curve obtained from NLFEA and experimental data is given in **Figure 23**.

**4.** The material parameters of DP criterion are expressed in terms of the cohesion and the internal friction angle of MC criterion in this study. Therefore, the surfaces of both DP and

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MC yield criteria are made to coincide along the compression meridian.

**Nomenclature**

*f*

*f*

*f*

*f*

*f*

*I*

*c* cohesion

*cmr* cohesion of mortar

*Em* modulus of elasticity

*hbr* height of the unit

*k* material constant

*α* material constant

*εmax* maximum tensile strain

*φ* internal friction angle

*φmr* internal f

*σ<sup>m</sup>* mean stress

*Ebl* modulus of elasticity of the blocks

*Emr* modulus of elasticity of the mortar

*bl* compressive strength of the blocks

*<sup>l</sup>* confining pressure exerted by FRP

*mr* compressive strength of the mortar

*p* vertical pre-compression of the wall

*mu* uniaxial compressive strength of the unit

<sup>1</sup> first invariant of the stress and the deviatoric tensor *J*<sup>2</sup> second invariant of the stress and the deviatoric tensor

*ξ* hydrostatic component of the current stress state

riction angle of mortar

*φ<sup>i</sup>* initial angle of internal friction in radians *φ<sup>f</sup>* final angle of internal friction in radians

*br* compressive strength of bricks

*hmr* height of the mortar thickness

*Ebr* modulus of elasticity of bricks

As shown in **Figure 24**, the analysis is terminated upon reaching the maximum tensile strain at a horizontal drift of 13.00 mm. Experimental value is approximately as 12.40 mm. The effect of tensile deformations grows up along the toe of the wall under increasing lateral load and the critical tensile strain is observed at the toe of the wall (**Figure 24**).

## **4. Conclusions**

This chapter deals with the implementation of the plasticity theory to URM and FRP-strengthened masonry walls under compression and shear by considering the constitutive behavior of masonry units, mortar, and FRP composite material separately. A DP-type yield criterion is also employed relating the cohesion and the internal friction angle of both unit and mortar to their uniaxial compressive strength by extending the previous approach to the FRP-strengthened masonry walls. For this purpose, a large parametric study is performed in order to explain the mechanical properties of unit and mortar with only a single parameter, that is, their uniaxial compressive strength [24]. In order to evaluate the accuracy of the proposed approach, the analytical results are compared with the experimental results of both unreinforced and strengthened masonry walls tested in two different experimental studies. Following conclusions are drawn:


**4.** The material parameters of DP criterion are expressed in terms of the cohesion and the internal friction angle of MC criterion in this study. Therefore, the surfaces of both DP and MC yield criteria are made to coincide along the compression meridian.
