A Frame Element Model for the Nonlinear Analysis of FRP-Strengthened Masonry Panels Subjected to In-Plane Loads

Abstract

A frame element model for evaluating the nonlinear response of unstrengthened and FRP-strengthened masonry panels subjected to in-plane vertical and lateral loads is presented. The proposed model, based on some assumptions concerning the constitutive behaviour of masonry and FRP material, considers the panel discretized in frame elements with geometrical and mechanical properties derived on the basis of the different states characterizing the sectional behaviour. The reliability of the proposed model is assessed by considering some experimental cases deduced from the literature.

1. Introduction

Masonry constructions certainly constitute an important part of the existing buildings in several countries and, in many cases, of their cultural heritage [1]. Past and recent seismic events [2, 3] have pointed out a significant level of vulnerability of such constructions and have posed the necessity to develop adequate and effective retrofit and strengthening systems able to improve their seismic resistance.

In the last decades, strengthening techniques based on the use of fiber-reinforced polymers (FRPs) materials have been proposed. Among these, one of the most commonly used technique is represented by FRP strips which can be externally applied on both the concrete masonry structures [4–6].

The literature related to FRP-strengthening systems has clearly demonstrated the capacity of such systems to improve the structural performances of masonry structures [7–9]. Some of the major drawbacks in modeling FRP-strengthened structures are certainly related to the interaction mechanism between masonry and FRPs at their interface layer. Many studies have specifically investigated the bond response of FRP applied on masonry supports [10–12], and some of them have provided theoretical laws for modeling such behavior and developing numerical models [13–20].

On the other side, some studies have focused on the definition of reliable numerical tools to be used in the assessment of the performances of the strengthened constructions and in the design of the strengthening systems themselves. Among these are the approaches developed within the finite element formulation framework. In particular, many authors have proposed modeling approaches based on the use of nonlinear interface elements interposed between the support and the FRP strengthening and characterized by constitutive laws derived by tests or by theoretical considerations [21, 22]. Other approaches have considered homogenization procedures for including the behavior of the FRP/support interface into the model of the FRP-reinforcement or, directly, in the masonry model [23–25]. In this context, the approaches based on the schematization of masonry elements strengthened with FRPs by using frame elements are of particular interest since they are in accordance with the equivalent frame approach which is useful in the design and implemented in many commercial computer codes for the structural analysis of masonry structures [26].

The model herein proposed for the nonlinear analysis of masonry panels strengthened with FRPs belongs to the equivalent frame approach and aims at providing a simple and effective tool to be used in the phase of assessment of masonry structures and, also, in the preliminary design of the FRP-strengthening systems.

2. The Proposed Model

The proposed model is based on simple considerations concerning the structural response of masonry panels subjected to in-plane loads.

Consider the masonry panel shown in Figure 1, subjected to a constant vertical load 𝑁 acting along the central axis of the panel and to a horizontal force 𝑉 applied at the top of the panel. By increasing the force 𝑉, the stress-strain state characterizing the behavior of the panel cross-sections varies continuously along its height. In particular, assuming for the masonry material a no-tensile behavior with an elastic perfectly-plastic response in compression, the following states can be identified (Figure 1):(a)the whole section is in compression, and the material behaves elastically (𝜀_max≤𝜀_𝑦);(b)the whole section is in compression, and the material behaves plastically (𝜀_max>𝜀_𝑦);(c)only a portion 𝑥 of the section is in compression, and here the material behaves elastically (𝜀_max≤𝜀_𝑦);(d)only a portion 𝑥 of the section is in compression, and here the material behaves plastically (𝜀_max>𝜀_𝑦);

Figure 1 

States characterizing the behavior of the cross-sections composing the panel.

with 𝜀_max being the maximum value of the strain in compression and 𝜀_𝑦 the strain value corresponding to the elastic limit of the masonry material.

At each value of the force 𝑉, the masonry panel can be considered as composed, along its height, of different homogenous zones with each one characterized by one of the above states. The number of these zones and their extension along the panel height depend on several parameters such as the masonry properties, the value of the applied loads, and the geometry of the panel.

The model proposed herein consists of schematizing the panel as an assemblage of elements, with flexure and shear deformability, representing the homogenous zones. An example of the model is reported in Figure 2. In particular, Figure 2(a) shows the case of a single element while Figures 2(b) and 2(c) show the case of more elements, one element at the top with all the sections fully compressed and the other elements where the sections are not entirely in compression.

Figure 2 

Unstrengthened panels: schemes of assemblage of frame elements.

The properties of each element—length and resisting cross-section—are defined by the properties of the corresponding homogenous zone. In particular, the cross-section area and the moment of inertia of each element are assumed to be constant and, for each step of load, are equal to the average value of the cross-section area and the moment of inertia characterizing the resisting sections located at the ends of the homogenous zone. Each element is, then, described by a stiffness matrix derived by taking into account also the shear deformability introduced through the terms 𝑠 and 𝑒 and given by𝐾_element=𝐸_𝑚𝐴𝐻012𝐸_𝑚𝐼𝑠𝐻³0sym.6𝐸_𝑚𝐼𝑠𝐻²4𝐸_𝑚𝐼(1+𝑒)−𝐸𝑠𝐻_𝑚𝐴𝐻𝐸00^𝑚𝐴𝐻0−12𝐸_𝑚𝐼𝑠𝐻³−6𝐸_𝑚𝐼𝑠𝐻²012𝐸_𝑚𝐼𝑠𝐻³06𝐸_𝑚𝐼𝑠𝐻²−2𝐸_𝑚𝐼(2𝑒−1)𝑠𝐻0−6𝐸_𝑚𝐼𝑠𝐻²4𝐸_𝑚𝐼(1+𝑒)𝑠𝐻, (1)

where 𝐻 is length of the considered element, 𝐴 average value of the cross-section area of the resisting sections located at the ends of the element, 𝐼 average value of the moment of inertia of the resisting sections,  𝐸_𝑚 Young’s modulus of masonry,  𝐺_𝑚 shear modulus of masonry;  𝑒=3𝐸_𝑚𝐼𝜒/𝐺_𝑚𝐴𝐻² with 𝜒 the shear factor of the resisting section and  𝑠=1+4𝑒.

Indeed, in the context of the equivalent frame modeling approach, the shear deformability plays a crucial role in deriving the seismic response of masonry structures since the capacity curve obtained through a pushover analysis is devoted to furnish both the peak load (resistance) of the structure and the failure mechanism affecting piers and spandrels, also a measure of the ultimate displacement capacity necessary for evaluating the global ductility of the structure.

Finally, the stiffness matrix of the panel 𝐾_panel is obtained by assembling the stiffness matrices of each element.

The model defined for the case of unstrengthened panels is, then, extended to the case of FRP-strengthened panels. In this case, the following additional hypotheses have been introduced:(i)a no-compression behavior for the FRP-strengthening system with a linear-elastic behavior in tension until its failure or the decohesion occurs;(ii)absence of slip phenomena between masonry and FRP-strengthening.

Considering the above hypotheses, it is evident that additional states, besides those (a) to (d) previously defined, would occur when the strengthening system is activated. Consequently, in these cases additional homogeneous zones will characterize the behavior of the FRP-strengthened panels.

The proposed model is applied to evaluate the nonlinear response of the panel in terms of its capacity curve. The procedure is articulated into two phases: the first phase aimed at analyzing the stress-strain state, and the second phase is devoted to evaluate the lateral displacement exhibited by the panel during the increments of the lateral force 𝑉.

The equations characterizing the first phase are equilibrium and constitutive equations and will be reported in detail in the following sections for unstrengthened and strengthened panels. These equations allow evaluating the stress-strain state at the base section of the panel, where deformations assume the maximum values, and at the sections where a change of the states occurs.

In the second phase, for each value of the applied force 𝑉 it is possible to derive the corresponding value of the lateral displacement 𝐷 at the top of the panel.

The capacity curve of the panel, that is, the curve in terms of the applied force 𝑉 versus the top displacement 𝐷, can be finally derived. This curve is representative of the global response of the panel and takes into account the process of formation of the homogenous zones characterizing the structural response of the panel.

The procedure is analyzed in detail in the following sections for unstrengthened and strengthened panels.

2.1. Unstrengthened Panels

With reference to the panel shown in Figure 2, subjected to a vertical force 𝑁, constant and acting along the central axis of the panel, and to a horizontal force 𝑉 monotonically increasing, the position of the neutral axis, indicated by 𝑥, defines two possible states at the base section of the panel:(a)the whole section is in compression (𝑥>𝐵, Figure 2(a));(b)only a portion of the section is in compression (𝑥<𝐵, Figures 2(b) and 2(c)).

In the first case, the panel is modeled through a single element (Figure 2(a)) while, in the second case, the panel is modeled by two or more elements (Figures 2(b) and 2(c)), one representing the portion of the panel where the sections are fully compressed and the others representating the parts where the sections are not fully compressed (𝑥<𝐵). For both cases, the attained maximum strain 𝜀_max activated in the sections cannot exceed the ultimate value 𝜀_𝑢 which indicates the failure condition for the panel. In both cases, the maximum strain can assume values less or greater than the yield strain 𝜀_𝑦; when 𝜀_max exceeds 𝜀_𝑦, three elements are necessary for schematizing the panel (Figure 2(c)).

The set of equations used for deriving the stress-strain state of the panel sections and the selected procedure for carrying out the capacity curve have been implemented in MATLAB [28]. The flowchart shown in Figure 3 schematically represents the main steps of the procedure for the case of unstrengthened panels. In particular, considering different positions of the neutral axis for the base section of the panel, the corresponding horizontal force 𝑉 and displacement 𝐷 at the top of the panel are derived by considering the following five steps of analysis.

Figure 3 

Flowchart of the procedure for unstrengthened panels.

(1)   Assign the position of the neutral axis, 𝑥, which could be external or internal to the section (𝑥≥𝐵 or 𝑥<𝐵, resp.).

(2) Evaluate the maximum strain in the masonry, 𝜀_max, through the following set of equations.

 Case of 𝑥≥𝐵. If 𝜀_max≤𝜀_𝑦, 𝜀_max=2𝑁𝐸_𝑚(),𝜀⋅𝐵⋅𝑡⋅1+((𝑥−𝐵/𝑥))_min=𝜀_max⋅𝑥−𝑏𝑥, (2) if 𝜀_𝑦<𝜀_max≤𝜀_𝑢, 𝜀_max𝑥=𝜀_min=𝜀𝑥−𝐵𝑥^𝑦_𝑠,𝑓_𝑚𝑘𝑥−𝑥_𝑠𝑓𝑡+^𝑚𝑘+𝐸_𝑚⋅𝜀_min2𝑡⋅2𝐵−2𝑥+𝑥_𝑠=𝑁. (3)

Case of 𝑥<𝐵. If 𝜀_max≤𝜀_𝑦, 𝜀_max=2𝑁𝐸_𝑚,⋅𝑥⋅𝑡 (4) if 𝜀_𝑦<𝜀_max≤𝜀_𝑢, 𝜀_max=(1/2)𝑓_𝑚𝑘⋅𝑡⋅𝑥⋅𝜀_𝑦𝑓_𝑚𝑘,⋅𝑡⋅𝑥−𝑁 (5) where 𝑓_𝑚𝑘 is the masonry strength and 𝑥_𝑠 indicates the fiber where the elastic limit strain is attained.

(3)   Check the derived value of 𝜀_max in order to establish the number of elements composing the panel.

(4) Evaluate the force 𝑉 corresponding to the assigned position of the neutral axis through the following set of equations which are dependent on the position of the neutral axis, the value of the maximum strain, and the characteristics of the elements composing the panel (number of elements, length and properties of the cross-section of each element).

 Case of 𝑥≥𝐵. If  𝜀_max≤𝜀_𝑦, )𝐸𝑉=(1/2_𝑚𝜀_max−𝜀_min⋅𝑡𝐵²/6𝐻, (6) if 𝜀_𝑦<𝜀_max≤𝜀_𝑢, 𝑌𝑉=¹+𝑌₂𝐻,𝑌₁=𝑓_𝑚𝑘+𝐸_𝑚𝜀_min𝑡𝑥−𝑥_𝑠𝐵−𝑥+𝑥_𝑠2,𝑌₂=12𝑓_𝑚𝑘−𝐸_𝑚𝜀_min𝑡2𝐵−2𝑥+𝑥_𝑠2𝑥−𝐵+𝑥_𝑠6. (7)

Case of 𝑥<𝐵. If 𝜀_max≤𝜀_𝑦, )𝐸𝑉=(1/2_𝑚𝜀_max𝑥⋅𝑡(𝐵/2−𝑥/3)𝐻,𝐻₁=𝐻−𝑁⋅𝐵,6𝑉 (8) if 𝜀_𝑦<𝜀_max≤𝜀_𝑢, )𝑓𝑉=(1/2_𝑚𝑘𝑥_𝑠⋅𝑡𝐵/2+2𝑥_𝑠/3−𝑥𝐻+𝑓_𝑚𝑘𝑥−𝑥_𝑠⋅𝑡𝐵−𝑥+𝑥_𝑠/2𝐻.𝐻₁=𝐻−𝑁⋅𝐵,𝐻6𝑉₂𝑁=𝐻−𝑉𝐵2−2𝑁3𝑓_𝑚𝑘.⋅𝑡 (9)

(5) Evaluate the top displacement 𝐷 of the panel corresponding to the force 𝑉 calculated at the previous step by using the stiffness matrix of the panel 𝐾_panel.

The procedure is then repeated starting from a new position of the neutral axis, and it continues until the ultimate value of the strain is attained. Then the capacity curve 𝑉-𝐷 of the panel is derived.

The capacity curve provides two pieces of information, the first related to the maximum horizontal force sustained by the panel and the second related to its ductility which is defined as the ratio between the ultimate displacement and the displacement corresponding to the attainment of the masonry yield strain.

It is important to emphasize that the different phases characterizing the behavior of the panel and, consequently, its schematization through frame elements require to operate in terms of increments of load since the nonlinearity of the response leads to a variation of the stiffness matrix of the model at each load increment.

2.2. FRP-Strengthened Panels

The model proposed for the case of the unstrengthened panels has been extended to the case of masonry panels strengthened with FRP strips bonded on the external surface (Figure 4). The equations used for deriving the capacity curve are those carried out for the case of the unstrengthened panels with additional equations referring to the sections where the strengthening is activated.

Figure 4 

Strengthening configuration of the panel.

In Figure 5 the flowchart schematizing the steps of the procedure for the case of FRP-strengthened panels is reported. It is possible to observe that the procedure differs with respect to the one developed for the unstrengthened panels only when the position of the neutral axis leads to the activation of the FRP strips. In Figures 6 and 7 the schemes of the strengthened panels used for deriving the equations are reported as follows.

Figure 5 

Flowchart of the procedure for FRP-strengthened panels.

Figure 6 

FRP-strengthened panels: schemes of assemblage of frame elements.

Figure 7 

FRP-strengthened panels: schemes of assemblage of frame elements.

(1)   Assign the position 𝑥 of the neutral axis.

(2) Evaluate the maximum value of the strain in the masonry and the strengthening through the following set of equations.

Case of 𝐵−𝑑₂≤𝑥<𝐵−𝑑₁. If 𝜀_max≤𝜀_𝑦, 𝜀_max𝑥=𝜀_𝑓1𝐵−𝑑₁,1−𝑥2𝐸_𝑚𝜀_max𝑡⋅𝑥−𝐸_𝑓𝜀_𝑓1𝐴_𝑓1=𝑁, (10) if 𝜀_𝑦<𝜀_max≤𝜀_𝑢, 𝜀_max𝑥=𝜀_𝑓1𝐵−𝑑₁=𝜀−𝑥𝑥^𝑦_𝑠,12𝑓_𝑚𝑘𝑡⋅𝑥_𝑠+𝑓_𝑚𝑘𝑡⋅𝑥−𝑥_𝑠−𝐸_𝑓𝜀_𝑓1𝐴_𝑓1=𝑁. (11)

Case of 0<𝑥≤𝐵−𝑑₂. If 𝜀_max≤𝜀_𝑦, 𝜀_max𝑥=𝜀_𝑓1𝐵−𝑑₁=𝜀−𝑥𝐵−𝑑^𝑓2₂,1−𝑥2𝜀_max𝐸_𝑚𝑡⋅𝑥−𝐸_𝑓𝜀_𝑓1𝐴_𝑓1−𝐸_𝑓𝜀_𝑓2𝐴_𝑓2=𝑁, (12) if 𝜀_𝑦<𝜀_max≤𝜀_𝑢, 𝜀_max𝑥=𝜀_𝑓1𝐵−𝑑₁=𝜀−𝑥𝐵−𝑑^𝑓2₂,1−𝑥𝑁=2𝑓_𝑚𝑘𝑡⋅𝑥_𝑠+𝑓_𝑚𝑘𝑡⋅𝑥−𝑥_𝑠+𝐸_𝑓𝜀_𝑓1𝐴_𝑓1+𝐸_𝑓𝜀_𝑓2𝐴_𝑓2, (13) where 𝜀_𝑓1 and 𝜀_𝑓2 are the tensile strains of the FRP strips at the base of the panel and 𝐴_𝑓1 and 𝐴_𝑓2 are the corresponding cross-section areas.

(3) Check the evaluated maximum strain in the masonry and FRP derived from the previous set of equations. In particular, the attainment of the ultimate deformation either in masonry or in FRP leads to stopping the procedure.

(4) Evaluate the force 𝑉 by introducing a further set of equations.

Case of 𝐵−𝑑₂≤𝑥<𝐵−𝑑₁. If 𝜀_max≤𝜀_𝑦, ()𝜀𝑉=1/2_max⋅𝐸_𝑚𝑡⋅𝑥(𝐵/2−𝑥/3)𝐻+𝐸_𝑓1𝜀_𝑓𝐴_𝑓𝐵/2−𝑑₁𝐻,𝐻₁=𝐻−𝑁⋅𝐵𝐻6⋅𝑉₂𝐵=𝐻−𝑁⋅6+𝑑₁31𝑉, (14) if 𝜀_𝑦≤𝜀_max≤𝜀_𝑢, 𝑌𝑉=¹+𝑌₂+𝑌₃𝐻,𝑌₁=12𝑓_𝑚𝑘𝑡⋅𝑥_𝑠𝐵2−𝑥−𝑥_𝑠−𝑥_𝑠3,𝑌₂=𝑓_𝑚𝑘𝑥−𝑥_𝑠𝐵𝑡⋅2−𝑥−𝑥_𝑠2,𝑌₃=𝜀_𝑓⋅𝐸_𝑓⋅𝐴_𝑓𝐵2−𝑑₁,𝐻₁=𝐻−𝑁⋅𝐵,𝐻6⋅𝑉₂𝐵=𝐻−𝑁⋅6+𝑑₁31𝑉,𝐻₃−()𝑓=𝐻1/2_𝑚𝑘𝑡⋅𝑥_𝑦𝐵/2−𝑥_𝑦/3𝑉,++𝜀_𝑓1,2𝐸_𝑓𝐴_𝑓1𝐵/2−𝑑₁𝑉,𝑥_𝑦=𝑁−𝜀_𝑓1,3𝐸_𝑓𝐴_𝑓1(1/2)𝑓_𝑚𝑘𝑡. (15)

Case of 0<𝑥≤𝐵−𝑑₂. If 𝜀_max≤𝜀_𝑦, 𝑌𝑉=¹+𝑌₂+𝑌₃𝐻𝑌₁=12𝜀_max⋅𝐸_𝑚𝐵𝑡⋅𝑥2−𝑥3𝑌₂=𝐸_𝑓𝜀_𝑓1𝐴_𝑓1𝐵2−𝑑₁,𝑌₃=𝐸_𝑓𝜀_𝑓2𝐴_𝑓2𝐵2−𝑑₂𝐻₁=𝐻−𝑁⋅𝐵𝐻6⋅𝑉₂𝐵=𝐻−𝑁⋅2−𝐵−𝑑₂31𝑉𝐻₃()𝑓=𝐻−1/2_𝑚𝑘𝑥_𝑦𝑡𝐵/2−𝑥_𝑦/3+𝜀_𝑓1,3𝐸_𝑓𝐴_𝑓𝐵/2−𝑑₁𝑉 (16) if 𝜀_𝑦≤𝜀_max≤𝜀_𝑢, 𝑌𝑉=¹+𝑌₂+𝑌₃+𝑌₄𝐻,𝑌₁=12𝑓_𝑚𝑘𝑡⋅𝑥_𝑠𝐵2−𝑥+2𝑥_𝑠3,𝑌₂=𝑓_𝑚𝑘𝑡⋅𝑥−𝑥_𝑠𝐵2−𝑥−𝑥_𝑠2,𝑌₃=𝐸_𝑓𝜀_𝑓1𝐴_𝑓1𝐵2−𝑑₁,𝑌₄=𝐸_𝑓𝜀_𝑓2𝐴_𝑓2𝐵2−𝑑₂,𝐻₁=𝐻−𝑁⋅𝐵,𝐻6⋅𝑉₂𝐵=𝐻−𝑁⋅6+𝑑₁31𝑉,𝐻₃()𝑓=𝐻−1/2_𝑚𝑘𝑡𝑥_𝑦𝐵/2−𝑥_𝑦/3𝑉+𝜀_𝑓1,3𝐴_𝑓1𝐸_𝑓𝐵/2−𝑑₁𝑉,𝐻₄𝑍=𝐻−¹+𝑍₂+𝑍₃𝑉,𝑍₁=12𝑓_𝑚𝑘𝑡⋅𝑥_𝑠4𝐵2−𝑥₄+2𝑥_𝑠43,𝑍₂=𝑓_𝑚𝑘𝑡⋅𝑥₄−𝑥_𝑠4𝐵2−𝑥₄−𝑥_𝑠42,𝑍₃=𝐸_𝑓𝜀_𝑓1,4𝐴_𝑓1𝐵2−𝑑₁. (17)

(5) Evaluate the top displacement 𝐷 of the panel corresponding to the force 𝑉 calculated at the previous step by using the stiffness matrix of the panel 𝐾_panel. Then the capacity curve 𝑉-𝐷 of the strengthened panel is derived. The capacity curve provides the maximum horizontal force sustained by the panel and its ductility defined as the ratio between the ultimate displacement, corresponding to either the masonry failure or the FRP failure/decohesion, and the displacement corresponding to the attainment of the masonry yield strain.

The process is then repeated starting from a new position of the neutral axis, and it continues until the ultimate value of the strain is attained.

3. Numerical Applications

The proposed model has been applied with reference to some cases of study deduced from the literature and for which the capacity curves were deduced experimentally.

The main characteristics of the analyzed panels are summarized in Table 1, and they refer to the following literature studies: Fantoni, 1981 [29], panels denoted by (a, b, and c); Giambanco et al. 1996 [30], panel denoted by panel (d); Giuffrè and Grimaldi, 1985 [31], panel denoted by panel (e); Callerio, 1998 [32], panel denoted by panel (f); Marcari, 2005 [27], panel denoted (g). The panels are characterized by different geometrical dimensions and mechanical properties of masonry, important for assessing the sensibility of the model.

Table 1 

Characteristics of the literature panels.

The reliability of the proposed model is analyzed, for each unstrengthened and strengthened panel, by comparing the capacity curve obtained by the model itself with the experimental curve. Further results have been also derived from the numerical analyses in order to better investigate the structural response of the panels.

3.1. Unstrengthened Panels

The capacity curves 𝑉-𝐷 deduced by using the proposed model developed for the unstrengthened masonry panels have been compared with the experimental curve derived from the related studies [29–32]. The comparisons are reported in Figures 8(a)–8(f) where the experimental results are indicated by circular symbols and the numerical results by continuous lines. Figure 8 shows a good agreement between the numerical and the experimental results in terms of the maximum load and, in many cases, in terms of ultimate displacement.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8 

Comparisons between numerical and experimental capacity curves of unstrengthened panels.

The proposed model is, then, applied to examine different aspects of the panels behavior. In Figure 9 the variation of the height 𝐻₂, is analyzed representing the amplitude of the plastic zone of the panel where the maximum strain exceeds the yield strain value of the masonry, as a function of the position of the neutral axis at the base section. From the plots of Figure 9 it is possible to observe a similar behavior for panels (a), (c), and (f) which are characterized by a maximum height 𝐻^∗₁ of the plastic zone less than 5% of the whole height of the panel and by a maximum base of the plastic zone less than 20% of its length 𝐵.

Figure 9 

Variation of the dimensions of the plastic zone for the unstrengthened panels.

For the other panels (b) and (d), although the height 𝐻^∗₁ of the plastic zone is less than 5% of the height of the panel, a greater portion of the base, about 30% of the base dimension, characterized the plastic zone.

In Figure 10 the curves representing the variation of the height of the different zones characterizing the behavior of panel (b) and panel (e) are reported: zone 1 with height equal to 𝐻^∗₁, where 𝜀_max>𝜀_𝑦; zone 2 with height equal to 𝐻^∗₂, where 𝜀_max<𝜀_𝑦, and only a portion of the section is in compression; zone 3 with height equal to 𝐻^∗₃, where 𝜀_max<𝜀_𝑦, and the whole section is in compression. In the figure the height 𝐻^∗_𝑖 is normalized with respect to the height 𝐻 of the panel and is plotted as a function of the position of the neutral axis at the base section.

Figure 10 

Variation of the dimensions of the zones characterizing the behavior of the panel.

The plots of Figure 10 clearly show the process of formation of the various zones characterizing the behavior of the panels when the external force 𝑉 increases. It is interesting to observe that in both cases 𝐻^∗₂=𝐻^∗₃ when the neutral axis is located at the center of the section (𝑥/𝐵=0.5), and the maximum value of the height 𝐻^∗₁ of the panel (b) is about 0.05𝐻 (i.e., 120 mm) and about 0.1𝐻 (i.e., 98 mm) for panel (e). The different amplitudes of the plastic zone affects the global response of the panels (Figure 8), where a greater plastic deformation for the panel (e) is evident.

3.2. FRP-Strengthened Panels

The comparison between the experimental and the numerical capacity curves deduced for the strengthened panel experimentally examined in [27] is shown in Figure 11. Also in this case the developed model shows a good reliability in reproducing the global response of the panel. Indeed, the numerical curve well approximates the prepeak behavior and provides a good estimation of the peak load. Nevertheless, a significant degradation characterizing the pre-peak experimental behavior of the panel probably due to the gradual detachment of the strips, which is not accounted by the proposed model is evident. Moreover, differently from the unstrengthened panels, the FRP-strengthened panel shows a significant softening in the prepeak phase which emphasizes a fragile response.

(a)

(b)

Figure 11 

(a) Dimensions and strips arrangement of the considered FRP-strengthened panel [27]. (b) Comparison between numerical and experimental capacity curves of the FRP-strengthened panel.

4. Conclusions

In this paper a simple approach for evaluating the nonlinear response of unstrengthened and FRP-strengthened masonry panels subjected to in-plane vertical and lateral loads has been presented. The model belongs to the equivalent frame approach and consists of schematizing the panel as an assemblage of frame elements with flexure and shear deformability. The geometrical and mechanical properties of the elements are derived on the basis of the different stress-strain states characterizing the sectional behaviour.

The reliability of the proposed model is analysed by considering some experimental cases deduced from the literature. The comparison in terms of the capacity curve obtained by the numerical and the experimental studies has provided a satisfactory agreement especially for the prediction of the maximum load sustained by the panels and, in most of the cases, in terms of the ultimate displacement. However, for some of the analysed unstrengthened panels and for the FRP-strengthened panel, the numerical values of the ultimate displacement were less than the experimental values. These results, certainly due to the simplified hypothesis assumed for the masonry and FRP and, in particular, for their interaction mechanism, have evidenced the restrictions of the proposed model and could constitute the basis from which further development of the model itself is proposed.

The model, as proposed herein and being based only on few parameters obtained by standard experimental tests on masonry and FRP, aims at providing a simple tool to be used in the phase of assessment of masonry structures and in the preliminary design of the FRP-strengthening systems. Moreover the model could be applied also to the case of building facades because it provides the capacity curves of each wall bay constituting the facades.

Symbols

𝑁:	Vertical load applied at the top of the panel along its central axis
𝑉:	Horizontal load applied at the top of the panel
𝐷:	Horizontal displacement at the top of the panel
𝐵:	Width of the panel
𝑡:	Thickness of the panel
𝐻:	Height of the panel
𝑑1:	Distance of the left FRP sheet from the panel edge
𝑑2:	Distance of the right FRP sheet from the panel edge
𝐴𝑓1, 𝐴𝑓2:	Cross-section areas of the FRP strips
𝐻𝑖(𝑖=1,…,4):	Height of the panel’s portion with sections not fully compressed
𝐻∗𝑖(𝑖=1,…,3):	Height of the homogenous zones of the panel
𝐸𝑚:	Young’s modulus of masonry
𝐺𝑚:	Shear modulus of masonry
𝑓𝑚𝑘:	Strength of masonry
𝜀𝑦:	Elastic limit strain of masonry
𝜀𝑢:	Ultimate strain of masonry
𝜀max:	Maximum strain of masonry in compression
𝜀min:	Minimum strain of masonry in compression
𝜀𝑓𝑢:	Ultimate strain of FRP
𝜀𝑓1:	Strain of the left FRP sheet at the base section
𝜀𝑓2:	Strain of the right FRP sheet at the base section
𝜀𝑓1,3:	Strain of the left FRP sheet at the section located at 𝐻3
𝜀𝑓1,4:	Strain of the left FRP sheet at the section located at 𝐻4
𝑥∶	Neutral axis of the section
𝑥𝑦:	Neutral axis of the section when 𝜀max=𝜀𝑦
𝑥𝑠:	Distance of the fiber when 𝜀𝑦 is attained
𝐾element:	Stiffness matrix of the single element
𝐾panel:	Stiffness matrix of the panel
𝐴:	Average value of the cross-section area of the resisting sections located at the ends of the element
𝐼:	Average value of the moment of inertia of the resisting sections
𝜒:	Shear factor of the resisting section.