Case studies

Table 33, TMS 402/602 (2016) and EN 1996-1-1 (2005) are the only ones that mention the shear (transverse) modulus, which should be estimated as 𝐺 = 0.4𝐸. Although not mentioned in these two standards, according to Croce et al. (2018), this relationship was obtained by applying Poisson's ratio (𝜐) equal to 0.25 in Equation 87, which correlates the elastic constants of an isotropic material.

𝐺 = 𝐸

2(1 + 𝜐) Eq. 87 Table 33: Longitudinal modulus of elasticity of masonry in different standards.

Standard Block type

Concrete Clay

ABNT NBR 16868-1 (2020)

800𝑓𝑚′ if 𝑓𝑏 ≤ 20 MPa 750𝑓𝑚′ if 𝑓𝑏 = 22 e 24 MPa

700𝑓_𝑚^′ if 𝑓_𝑏 ≥ 26 MPa

600𝑓𝑚′

TMS 402/602 (2016) 900𝑓𝑚′ 700𝑓𝑚′

CSA S304 (2014) 850𝑓_𝑚^′ ≤ 20 GPa 850𝑓_𝑚^′ ≤ 20 GPa

EN1996-1-1 (2005) 1000𝑓𝑚′ 1000𝑓𝑚′

NZS 4230 (2004) 15 GPa 15 GPa

Source: Author.

Figure 99(c), it was recommended to include the axial and shear deformations in the analysis when using computational tools that help the calculations. The equivalent frame had its behavior changed when considering the horizontal links connecting the elements with infinite stiffness, Figure 99(d). In the continuum method, Figure 99(e), the connection between the walls was replaced by an equivalent shear medium continuous over the full height of the walls, not considering the axial deformation of the medium and the shear deformation of the walls.

According to the authors, the finite element method, Figure 99(f), is a powerful analysis tool for complex structures but not very practical for the usual design situations at that time.

Figure 98: Experimental structure of reference for the study of Hendry (1981).

Source: Hendry (1981).

Figure 99: Theoretical idealization of walls with openings by Hendry (1981).

(a) Elevation (b) Individual cantilever (c) Equivalent frame

(d) Wide column frame (e) Continuum (f) Finite elements Source: Hendry (1981).

Hendry (1981) performed the analyses by adapting the three-dimensional system to an equivalent two-dimensional one of walls and beams with the same areas and moments of inertia of the actual structure. The comparison between the lateral displacements is shown in Figure 100(a), while the stress distribution along the base of the walls at the first story for the experiment and models can be seen in Figure 100(b).

Figure 100: Comparison of the results of the models evaluated by Hendry (1981).

(a) Deflection (b) Stress at wall base Source: Adapted from Hendry (1981).

The results showed that the models closest to the structure's actual behavior were the equivalent frame and the finite element, the latter being the only one capable of simulating the non-linear stress distribution. The wide column frame and continuum methods did not present satisfactory results; therefore, their use was not recommended by the authors. The cantilever method provided very conservative results and, according to the authors, should be used for preliminary estimations of bending moments and shear forces on the walls. It was also commented that the hypotheses assumed for the interaction between the elements in the models may have caused the differences between the experimental and theoretical results.

b) Kappos et al. (2002)

Kappos et al. (2002) evaluated the accuracy of models for practical use in engineering to analyze unreinforced masonry buildings subjected to lateral actions. The authors' main objective was to determine under which conditions the equivalent frame model can be used to design and verify masonry structures. The study was initially carried out in an elastic regime for a typical wall with openings and a real three-dimensional building with and without the rigid diaphragm effect provided by the slabs; then, analyzes were made for the inelastic behavior of the typical wall. The wall in question refers to previous studies of Seible and Kingsley (1991), while the three-dimensional structure was discussed by Karantoni and Fardis (1992).

The authors tested several variations of the equivalent frame (EF) and plane finite element (FE) models for the typical wall. Concerning the equivalent frame model, the options were: full rigid offsets horizontally (EF1), full rigid offsets horizontally and vertically (EF2), Figure 101(a), and full horizontal and half vertical rigid offsets (EF3). This last consideration was thought to clarify the doubt about the extension to be adopted for the rigid offsets since its excess can confer a much greater stiffness to the model than exists in the actual wall. Variations in the plane finite element model are related to mesh refinement; a model with a coarse mesh was generated using elements with dimensions delimited by the geometry of the walls (FE-C), Figure 101(b), and another model with a smaller mesh (FE-R), Figure 101(c). Versions with rigid diaphragm were designed for all models (EF1D, EF2D, EF3D, FE-CD, and FE-RD). Such modeling was extended to the three-dimensional analysis of the building.

Concerning the material properties, Kappos et al. (2002) considered the longitudinal modulus of elasticity of masonry according to the European standard recommendation, 𝐸 = 1000𝑓_𝑚^′, maximum compressive strain 𝜀_𝑢 = 0.002 and tensile strength 𝑓_𝑡 = 0.1𝑓_𝑚^′. For the nonlinear analyses, a constitutive model was adopted with a parabolic stress-strain relationship

and stiffness degradation due to cracking when the element presented a compression deformation higher than 𝜀_𝑢. After reaching the cracking condition, the residual shear stiffness was recalculated as 60% of the uncracked value. An additional analysis was performed with a reduced value for the modulus of elasticity, 𝐸 = 550𝑓_𝑚^′, which showed an improvement in the behavior of the initial branch of the numerical curve compared to the experimental curve.

Figure 101: Wall geometry and models analyzed by Kappos et al. (2002).

(a) EF with full rigid offsets (b) FE with coarse mesh (c) FE with smaller mesh Source: Kappos et al. (2002).

The main discussion of results was about the displacements at the story levels, which are presented in the graphs of Figure 102. The authors concluded that the effect of the rigid diaphragm was negligible in the models of the 2D structure, Figure 102(a), but it was crucial for the general behavior of the 3D structure, Figure 102(b), distributing the displacements between the shear walls and not causing significant out-of-plane deformations. The mesh refinement did not make any difference concerning the displacements; however, it was essential in mapping the distribution and concentration of stresses. The equivalent frame model with full horizontal and vertical rigid offsets was the frame option that presented elastic displacements consistent with the refined mesh finite element model and stress results with differences within an acceptable range for design. Furthermore, the inelastic analyses showed that the response of these two models for the initial stiffness and the lateral load capacity of the wall were similar.

It was impossible to establish an inelastic comparison of the ultimate displacements because the finite element model was induced by forces while the equivalent frame model was induced by displacements.

Figure 102: Comparison of displacements of the models tested by Kappos et al. (2002).

(a) Plane structure (typical wall)

(b) Three-dimensional structure (building) Source: Kappos et al. (2002).

c) Tena-Colunga and Rivera-Hernandez (2018), Tena-Colunga and Liga-Paredes (2020)

These authors proposed a geometric adaptation to the equivalent frame and continuum models in situations with several openings in the same story, especially when misaligned. The focus of the new approach was to obtain, specifically, in a simplified and approximated way, the lateral stiffness and the profile of elastic lateral displacements of shear walls with multiple openings since these parameters are essential in displacement-based design methods.

The proposed simplification consists of transforming a wall with multiple openings into an equivalent wall with a single opening. The fundamental premise defended by the authors is that if the area and the effective eccentricity of the openings are kept constant, the stiffness and lateral displacements of the equivalent wall will be similar to those of the actual wall. Some adjustments are indicated for the geometric properties of the cross-section, adopting, then, equivalent area and second moment of area.

Walls with different arrangements of openings and number of floors (3, 6, 12, and 18) were evaluated considering the proposed simplification attributed to equivalent frame models,

continuum, and plane finite elements with refined mesh, the latter adopted as the reference.

Three variations were considered for the equivalent frame model: one version assuming perfect rigid offsets (infinite area and inertia) at the ends of the vertical and horizontal bars, Figure 103(a); another version considering perfect rigid offsets only at the end of the horizontal bars;

and the last one with rigid offsets with the equivalent area and inertias calculated as a function of the wall geometry, Figure 103(b), as proposed by Schwaighofer and Microys (1969) and described in Equations 88 to 91.

Figure 103: Variations adopted by Tena-Colunga and Rivera-Hernández (2018) in the rigid offsets of the equivalent frame model.

(a) Horizontal and vertical rigid offsets (b) Horizontal rigid offsets Source: Tena-Colunga and Rivera-Hernandez (2018).

𝐴_𝑒 = 𝐾₁𝐴_𝑓 Eq. 88 𝐼_𝑒 = 𝐾₂𝐼_𝑓 Eq. 89

𝐾₁ = 100 (^𝑒

𝑓) Eq. 90 𝐾₂ = 0.0593 (^𝑒

𝑓)⁴+ 99.348 (^𝑒

𝑓)³+ 302.43 (^𝑒

𝑓)²+ 296 (^𝑒

𝑓) + 1.7778 Eq. 91 where: 𝑒 is the distance between the vertical bar and the opening side, and 𝑓 is the distance between the opening side and its central axis, see Figure 103(b).

The authors commented that the gain in wall stiffness conferred by the slabs can be incorporated into the frame models through the inertia admitted for the coupling beam. Based on the section transformation principle, an equivalent thickness (𝑏_𝑒𝑞) is calculated for the slab

as a function of the extension of its flanges using Equation 92. Thus, the coupling beam has its second moment of area calculated for the geometry highlighted in Figure 104.

𝑏_𝑒𝑞 = 𝑏_𝑓(^𝐸𝑠

𝐸_𝑤) Eq. 92 where: 𝑏_𝑓 is the extension of the flanges, generally defined as a function of the slab thickness;

𝐸_𝑠 and 𝐸_𝑤 are, respectively, the longitudinal modulus of elasticity of the slab and wall.

Figure 104: Equivalent geometry for the coupling beams considering the slabs.

(a) Intermediate stories (b) Top story Source: Tena-Colunga and Liga-Paredes (2020).

The results were plotted graphically in displacement profiles similar to those in Figure 102 for the different situations evaluated, including model variations and opening parameters for the proposed simplification. Regarding the frame models, the analysis allowed to conclude that, in general, the version with the rigid offsets suggested by Schwaighofer and Microys (1969) presented the best approximations. In addition, the contribution of the slab stiffness was more significant for walls with typical door openings and less impacting on walls with symmetrically arranged window openings since, in this situation, stiffer beams (greater height) were already formed.

d) Pirsaheb et al. (2021)

Pirsaheb et al. (2021) proposed a numerical procedure to assess the progressive failure of unreinforced perforated masonry shear walls. The method, called the Multi-Pier-Macro (MPM) method, consists of equivalent braced frames simulating the piers and spandrels, as illustrated in Figure 105. The diagonals (braces) were set as truss elements with elastic and plastic deformation, while the vertical and horizontal elements were set as beam elements with elastic flexural behavior and with inertia sufficiently small to maintain the behavior mostly axially. The inelastic behavior was computed using concentrated tensile and compressive

plastic hinges, which their color indicates the level of plasticization according to the stress- strain curve adopted as the material constitutive relationship.

The geometrical dimensions of the vertical and horizontal elements were calculated by equating the second moment of area of the equivalent system and the actual wall. In turn, the cross-section area of braces was calculated by equating the shear stiffness of the actual wall and the equivalent system. Diagonals axially rigid were used in the intersection regions between piers and spandrels to represent the rigid zones.

Figure 105: Equivalent geometry for the coupling beams considering the slabs.

(a) Full-scale masonry wall (b) MPM equivalent system Source: Pirsaheb et al. (2021).

The failure mechanism is deduced based on which plastic hinge is activated. Horizontal and vertical tensile cracks and crushing occur when the hinges in the vertical and horizontal elements on piers and spandrels yield. Shear failure and toe crushing are identified when the plastic hinges in the braces reach their limits.

The authors stated that the approach can be useful in typical design for its simplicity.

Also, the authors highlighted the fast execution, the reliability in simulating the global behavior and damage progress, and the need for only generalist FE software with non-linear trusses.

5.2 LINEAR FRAME MODELS