DATABASE

Koutras and Shing (2018) commented that the equations of the American standard predicted the shear capacity of the walls of their experiment sufficiently well considering the sum of the capacity of the piers. However, the disparity between the stiffnesses of the piers and their brittle behavior can lead to an unsafe design.

Table 24: Properties of materials used in the improved modeling.

Material Parameter Value for the material type Reference of the used value

(a) (b)

Concrete (a) Middle slabs

(b) Top beam

Compressive

Strength (𝑓_𝑐^′) 36.1 MPa 31.5 MPa Test results

(Fortes and Parsekian, 2017) Tensile

Strength (𝑓_𝑡^′) 1.98 MPa 1.85 MPa 0.33√𝑓𝑐′

(Wong et al., 2013) Initial Tangent

Modulus (𝐸_𝑐) 33,046 MPa 30,869 MPa 5500√𝑓𝑐′

(Wong et al., 2013) Strain at Peak

Stress (𝜀_𝑝) 2.07 ∙ 10⁻³ 2.04 ∙ 10⁻³ 1.8 + 0.0075𝑓_𝑐^′ (Wong et al., 2013) Poisson’s

ratio (𝜈) 0.15 0.15 Default value

(Wong et al., 2013)

Masonry (a) Ungrouted

(b) Grouted

Compressive

Strength (𝑓𝑚′) 11.8 MPa 12.2 MPa Test results

(Fortes and Parsekian, 2017) Tensile

Strength (𝑓𝑡′) 1.13 MPa 1.15 MPa 0.33√𝑓𝑚′

(Wong et al., 2013) Joint shear

strength (𝑓𝑗) 0.37 MPa 0.37 MPa Test results

(Pasquantonio et al., 2020) Initial Tangent

Modulus (𝐸_𝑚) 18,305 MPa 18,496 MPa 3320√𝑓𝑚′ + 6900

(Wong et al., 2013) Strain at Peak

Stress (𝜀_𝑝) 1.29 ∙ 10⁻³ 1.32 ∙ 10⁻³ (2000𝑓_𝑚^′) 𝐸⁄ _𝑚 (Wong et al., 2013) Poisson’s

ratio (𝜈) 0.15 0.15 Default value

(Wong et al., 2013) Reinforcement

(a) Ø9.5 mm (b) Ø4.2 mm

Yield

Strength (𝑓𝑦) 540 MPa 743 MPa Test results

(Fortes and Parsekian, 2017) Ultimate

Strength (𝑓𝑢) 742 MPa 812 MPa Test results

(Fortes and Parsekian, 2017) Elastic

Modulus (𝐸_𝑠) 203,512 MPa 222,799 MPa Test results

(Fortes and Parsekian, 2017) Source: Author.

Figure 53: Experimental and numerical response of walls W1, W2, D1 and D2.

(a) Hysteresis curves of wall W1 (b) Hysteresis curves of wall W2

-0.72 -0.48 -0.24 0.00 0.24 0.48 0.72

-120 -90 -60 -30 0 30 60 90 120

-30 -20 -10 0 10 20 30

Top Drift (%)

Lateral Load (kN)

Lateral Displacement (mm) Experimental FE Model

-0.72 -0.48 -0.24 0.00 0.24 0.48 0.72

-120 -90 -60 -30 0 30 60 90 120

-30 -20 -10 0 10 20 30

Top Drift (%)

Lateral Load (kN)

Lateral Displacement (mm) Experimental FE Model

(c) Hysteresis curves of wall D1 (d) Hysteresis curves of wall D2

(e) Backbone curves of walls W1 and W2 (f) Backbone curves of walls D1 and D2 Source: Author.

As can be seen in Figure 53, there was good agreement between the backbone curves of the improved FE models and the experimental walls, with remarkable enhancement in capturing the strength and stiffness degradation at the post-peak stage when compared to the oldest models. Looking at the hysteresis curves, the improved model simulated the loops well, mainly up to the peak load, although some non-agreements are still noted in the post-peak stages, especially for the residual deformations. As explained by Elmeligy et al. (2021), these divergences can be attributed to brittleness and significant anisotropy in the ungrouted parts, which induces some randomness to the post-peak response. This explanation is reinforced by the differences observed when comparing the hysteresis of the experimental walls W1 and W2, and D1 and D2, which are nominally identical but have different post-peak responses.

-0.72 -0.48 -0.24 0.00 0.24 0.48 0.72

-120 -90 -60 -30 0 30 60 90 120

-30 -20 -10 0 10 20 30

Top Drift (%)

Lateral Load (kN)

Lateral Displacement (mm) Experimental FE Model

-0.72 -0.48 -0.24 0.00 0.24 0.48 0.72

-120 -90 -60 -30 0 30 60 90 120

-30 -20 -10 0 10 20 30

Top Drift (%)

Lateral Load (kN)

Lateral Displacement (mm) Experimental FE Model

-0.72 -0.48 -0.24 0.00 0.24 0.48 0.72

-120 -90 -60 -30 0 30 60 90 120

-30 -20 -10 0 10 20 30

Top Drift (%)

Lateral Load (kN)

Lateral Displacement (mm) Exp. Wall W1 Exp. Wall W2 FE Model (New) FE Model (Old)

-0.72 -0.48 -0.24 0.00 0.24 0.48 0.72

-120 -90 -60 -30 0 30 60 90 120

-30 -20 -10 0 10 20 30

Top Drift (%)

Lateral Load (kN)

Lateral Displacement (mm) Exp. Wall D1 Exp. Wall D2 FE Model (New) FE Model (Old)

The average experimental maximum lateral load and the average corresponding lateral displacement of walls W1-2 were 99.3 kN and 11.1 mm, respectively. The FE model for these walls resulted in the average peak load of 99.5 kN and the average displacement of 10.8 mm, which were 0.2% higher and 2.6% smaller, respectively than the experimental results. For walls D1-2, the experimental results for the average peak load and the average corresponding displacement were 98.9 kN and 16.2 mm, respectively, against 98.8 kN (-0.1%) and 14.9 mm (-8.0%) obtained from the numerical models.

The experimental walls failed predominantly in shear with diagonal tensile cracks. This behavior was also observed in the improved numerical models in a similar pattern to that discussed previously in section 2.3.2 and illustrated in Figure 33 and Figure 34. Therefore, after comparing the lateral load capacity, lateral displacement at the top of the wall, and crack pattern against the data of the experimental tests, the improved FE model can be considered able to represent the walls’ responses adequately.

After validation, other ninety-six walls were modeled aiming to assess the prediction of the SLC of PGMW varying the parameters included in the existing shear expressions or for which their influence is still questionable by researchers. Taking the experimental walls as the base, the simulated walls had changes in parameters such as the masonry compressive strength, the vertical and horizontal grouting and reinforcement, the applied axial load, the wall aspect ratio, and the dimensions of openings. A detailed list of the simulated walls is presented in Table 41 in Appendix C.

The nominal flexural load capacity of all walls was calculated to ensure shear failure.

The calculation was done as explained in section 3.1.1, including the contributions of the compression reinforcement and the weight of the walls, accounted for with the axial load. Even in the worst case, the nominal flexural load capacity was at least 43% higher than the maximum lateral load reached for that wall. In addition, the failure mode of all walls was examined and confirmed to be a shear failure because of the diagonal shear cracks, the absence of the vertical reinforcement yielding, and because toe crushing was not observed.

The reduced-scale numerical walls were converted to equivalent full-scale walls to proceed with the analyses and to be used in the assessment of the shear strength prediction equations. The simple model similarity approach (Tomaževič and Velechovsky, 1992) was employed, in which the geometric properties are scaled by a factor of SL (the ratio of the reduced-scale to the equivalent full-scale size), areas and forces are scaled by a factor of SL²,

and the material strengths are scaled by a factor of 1 (Long, 2006; Dillon and Fonseca, 2015;

Izquierdo et al., 2021). Taking the dimensions in the length for a block plus a joint in the reduced-scale as 185 + 5 = 190 mm, and in the full-scale as 390 + 10 = 400 mm, the factor SL

(190/400) = 0.475 for the walls in this study.

3.2.2 Experimental walls from the literature

Some criteria were established for the data collection from tests reported in the literature to avoid inconsistencies and high variation. The scope was restricted to masonry walls made of concrete blocks, partially grouted, subjected to in-plane reverse cyclic load or phased- sequential, quasi-static loading rate, which failed in a shear mode. Since the first numerical walls modeled considered half-scale walls experimental data, another criterion was to choose experimental walls constructed with full-scale units.

The data reported by Meli et al., (1968), Schultz (1996), Minaie et al. (2010), Elmapruk and ElGawady (2010), Nolph and ElGawady (2012), Hoque and Lissel (2013), and Rizaee and Lissel (2015) were selected to compose the experimental database. The data were extracted from the respective research and checked in the database assembled by Dillon and Fonseca (2015), and Izquierdo and Cruz-Noguez (2021). A detailed list of the selected walls is presented in Table 42 in Appendix C.

3.3 SHEAR LOAD CAPACITY PREDICTION OF UNPERFORATED PARTIALLY