Conceptualization

An axial load of 50 kN, which corresponds to approximately 14 kN/m, representing a typical service load for a 3-floor residential building, was uniformly distributed on the top of the walls. The load was applied using a steel section positioned over the walls. A neoprene pad was placed between the steel section and walls to minimize points of stress concentrations.

The horizontal load was applied to the top of the walls in the form of quasi-static, displacement-controlled cyclic loads with two complete repetitions of pushing and pulling per cycle. As shown in Figure 30, a total of 37 displacement cycles were planned in four phases: 7 initial cycles with an increment of 0.8 mm, 11 cycles with an increment of 2.0 mm, 13 cycles with an increment of 3.2 mm, and 6 cycles with an increment of 4.8 mm. The tests were stopped when the lateral force dropped to approximately 60% below the maximum measured force.

Figure 30: Actuator load protocol.

Source: Chavez and Fonseca (2018).

Plane membrane elements were employed with dimensions of 95 × 95 mm to represent each cell of the blocks, an approached that has been used by others (Lotfi and Shing, 1991;

Haach et al., 2011; Facconi et al., 2014; Elmeligy et al., 2021). The thickness was adopted equal to 90 mm for the grouted elements and 30 mm for the ungrouted elements, which is equivalent to the sum of the thickness of face shells. Blocks, concrete, and reinforcement in masonry were modeled as discrete elements whereas grout, mortar and reinforcement in concrete were considered smeared with the blocks and beam elements. The elements were connected directly node by node, and as such, the displacements of the elements were compatible. The mesh and model configuration of walls W1 and W2 are illustrated in Figure 31.

Figure 31: Numerical arrangement of walls W1 and W2.

Source: Author.

Since the tensile strength of the masonry was not determined experimentally by Fortes and Parsekian (2017) and it is generally low, it was assumed to be zero, which is conservative.

The ungrouted and grouted compressive masonry strengths were set equal to 11.8 and 12.2 MPa, according to Table 2. The initial tangent elastic modulus of the ungrouted and grouted masonry were not measured experimentally; rather, they were calculated using Equation 1 (Wong et al., 2013; VecTor2, V4.4, 2019), resulting in 18,305 and 18,496 MPa.

𝐸_𝑚 = 3320√𝑓𝑚′ + 6900 𝑀𝑃𝑎 Eq. 1 The mortar shear strength was also not obtained experimentally but was estimated as 265 kPa using Equation 2 (TMS 402/602, 2016). PGMW typically fail through the ungrouted

masonry joints (Shing and Cao, 1997; Drysdale et al., 1999); thus, Equation 2 was used with the compressive strength of ungrouted masonry as a substitute for mortar shear strength. In the software, the mortar shear strength is considered indirectly through the parameter joint shear strength ratio (JSSR), which is the ratio between the mortar shear strength and the masonry compressive strength. A weighted average of the grouted and ungrouted masonry compressive strengths was considered to estimate the joint shear strength ratio, resulting in a value of 0.0189.

𝑉_𝑛 = 0.083𝛾_𝑔[4.0 − 1.75 ( ^𝑀

V𝑑_𝑣)] 𝐴_𝑒ℎ√𝑓_𝑚^′ Eq. 2 The masonry is treated in the software as an orthotropic continuum material with joint failures smeared across the finite element and controlled by the smeared crack approach. Even when this continuum is uncracked, it may slip at the head and bed joints in a single finite element since the DSFM was adjusted for masonry materials (Wong et al., 2013; VecTor2, V4.4, 2019). The mesh is thus conditioned to the block size. The user should specify the spacing between head and bed joints, which in this case were 190 and 95 mm in the x and y directions in agreement with the physical walls. As suggested by Vecchio and Lai (2004), the Walraven stress model (Walraven, 1981; Walraven and Reinhardt, 1981), which is based on an analysis of the crack structure and contact area of crack faces, was adopted to crack slip calculations.

The yield and ultimate strengths, and the elastic modulus of the reinforcement, used both in masonry and in concrete beams, were adopted in the models according to the test results shown in Table 1. The reinforcement was categorized as ductile steel with a trilinear stress- strain response: an initial linear-elastic response, a yield plateau, and a linear strain-hardening phase until rupture. The hysteretic response, dowel action, and buckling effects were also incorporated in the models (Wong et al., 2013; VecTor2, V4.4, 2019).

The axial load was applied as 1.34 kN at the nodes along the wall top, for a total of 50.7 kN. The self-weight was also considered using 2,400 kg/m³ and 2,250 kg/m³ as the mass density for the concrete and masonry. The lateral displacement was applied using a cyclic incremental factor of 2.65 mm divided into 0.2 mm steps, which was calculated using a weighted average of the number of cycles and displacements.

The model of Hognestad (Hognestad, 1951) and the Base Curve model (Palermo and Vecchio, 2001) were adopted for pre-peak and post-peak compression responses, respectively.

The stress-strain curve of the Hognestad model is a parabola described by Equation 3 with a symmetric relationship at peak stress corresponding to 𝜀_𝑝 strain, decreasing to zero stress at

zero and 2𝜀_𝑝 strain points. This model is suitable for concrete with a compressive strength of less than 40 MPa, and as the masonry had a compressive strength not beyond this value, the model was considered a proper choice. The Base Curve model for the post-peak compression phase is a coherent choice if the Hognestad model is used for the pre-peak compression phase.

With this model, the post-peak compressive stresses are computed using the equations of the descending part of the adopted stress-strain curve (Wong et al., 2013; VecTor2, V4.4, 2019).

𝑓_𝑐𝑖 = −𝑓_𝑝[2 (^𝜀𝑐𝑖

𝜀𝑝) − (^𝜀𝑐𝑖

𝜀𝑝)

2

] < 0 𝑓𝑜𝑟 𝜀_𝑐𝑖 < 0 Eq. 3

The model proposed by Vecchio (Vecchio, 1999) was adopted to define the hysteretic response accounting for internal damage with plastic offsets and nonlinear loading/unloading (Wong et al., 2013; VecTor2, V4.4, 2019; Elmeligy et al., 2021). The model uses the failure criterion for masonry established by Ganz (Ganz, 1985) in terms of the principal stresses and the Mohr-Coulomb Stress as the cracking criterion with a cohesion considering the friction angle φ as 37° (Wong et al., 2013; Angelillo et al., 2014; Abdulla et al., 2017; VecTor2, V4.4, 2019; Elmeligy et al., 2021). The analysis involved static-nonlinear load steps with the convergence criterion being the weighted average of the displacements with a convergence limit equal to 1.00001 or 50 iterations. At each load step, the stiffness of the structure was recalculated using the mathematical constitutive models based on stresses and strains. More specific information about the constitutive models can be found in Appendix A.