LIST OF SYMBOLS
3. SHEAR LOAD CAPACITY PREDICTION
3.4 SHEAR LOAD CAPACITY OF PERFORATED PARTIALLY GROUTED MASONRY WALLS
grout only at the ends (walls 10-14), being the error higher as long as the length of the wall.
Contradistinctively, this equation predicted highly conservative values for the SLC of walls with lower spacing between grouted cores (walls 1, 2, 5, 7-9, 24, 25, 29, 30) when compared to other walls in the same subset.
Among the existing shear equations, the best estimations for the SLC of the experimental walls were made using the equations of Izquierdo et al. (2021) and Seif ElDin et al. (2019a). These equations estimated similarly the shear capacity of the walls with an average ratio Vn,pred/Vmax,avg close to 1.0 with a CoV of approximately 22%. The values for the 5th and 95th percentiles were approximately 0.75 and 1.5, respectively, and the RMSE was approximately 45 kN. Using these two equations, the most unsafe predictions were for walls 3, 4, and 6, which had a large spacing between grouted cores.
As with the numerical dataset, the proposed equation presented the best statistical indicators among all studied shear equations for the experimental dataset. The SLC of the walls was estimated with an average ratio Vn,pred/Vmax,avg of 1.05 with a CoV of 14%, and with the smallest range between the 5th percentile (0.81) and the 95th percentile (1.28). The ME of -6.0 kN and the RMSE of 33.9 kN for the proposed equation were also the lowest errors among all equations examined.
3.4 SHEAR LOAD CAPACITY OF PERFORATED PARTIALLY GROUTED
and 88: 2%, 8%, and 8%, respectively. The reduction in the Vmax,avg was smaller for walls with more stories probably because the diagonal struts found more regions of masonry between the stories to spread out, reducing relatively the influence of the openings. It is important to mention that the lateral load was applied at the top of the highest story; thus, the main diagonal strut starts from the point of application of the load and goes through the stories until the base of the wall. If the lateral load was applied at the level of each story, struts would form at each story, which would probably increase the influence of the openings.
Variations on the height of the openings also impacted the SLC of the walls. Decreases in Vmax,avg of approximately 6%, 9%, and 7% are observed when comparing respectively, walls 59 and 64, 60 and 65, and 61 and 66. Walls 59, 60 and 61 were single-story with six course- high window openings whereas walls 64, 65, and 66 were the same but with 11 course-high door openings. Making the same comparison for the two-story walls 75 and 78, 76 and 79, and 77 and 80, the Vmax,avg decreased by 7%, 11%, and 15%, respectively. For the three-story walls (86 and 89, 87 and 90, and 88 and 91) the Vmax,avg decreased 4%, 5%, and 15%, respectively. It appears that as the height of the openings increases, the wall panel behavior becomes more independent and, consequently, the diagonal struts concentrate in the shorter piers.
3.4.1 Examined approaches for predicting the shear capacity
Current shear equations consider only walls without openings: thus, four different approaches using the new proposed equation were examined to predict the SLC of perforated walls. The approaches are based on the strength of the wall piers with dimensions limited by different possibilities for the diagonal shear cracks.
The first approach (SA1) considers the full wall with a diagonal shear crack formed from the top of the bond beam of the last story to the base of the first course of the first story, Figure 59(a). The horizontal effective cross-sectional area is subtracted from the cross-sectional area of the openings.
In the second (SA2), third (SA3), and fourth (SA4) approaches, the openings are assumed to separate the wall into identical panels, rigidly connected and with the same strength.
The distinction between these approaches is where the diagonal shear crack is assumed to start and finish in the panels, which implies different dimensions for them. SA2 considers the diagonal shear crack to form from the top of the bond beam of each story to the base of the first course of the same story, Figure 59(b). In SA3, the diagonal crack is assumed to run from the top of the bond beam of each story to the bottom corner of the opening of the same story,
Figure 59(c). In SA4 the diagonal crack is taken to form from the upper corner of the opening to the lower corner of the opening of the same story, Figure 59(d).
Figure 59: Layout types that are considered on the four approaches.
(a) SA1 (b) SA2
(c) SA3 (d) SA4 Source: Author.
With approaches SA2, SA3 and SA4 the lateral load capacity is calculated as the sum of the capacities of the piers placed at the same horizontal alignment since it is the direction of the shear action. The sum of the capacities of the piers superposed vertically must not be included since this will overestimate the actual capacity of the wall. The most unfavorable case can be considered for the piers in the first story, and, therefore, the total axial load should be
divided between them in the calculation of the lateral load capacity. Only the reinforcement included in the piers was considered in the calculations using these approaches.
In addition to walls with only an opening positioned symmetrically in each story, the assessed dataset (walls 59 to 96 in Table 41) also included walls with two identical openings positioned symmetrically in each story (walls 61, 66, 77, 80, 88, and 91) and walls with two different openings positioned symmetrically in each story (walls 67, 81, and 92). The definition of the piers for these walls using the approaches SA3 and SA4 is conditioned by the dimension of the smallest opening.
3.4.2 Accuracy of the approaches for predicting the shear capacity
The results of the four approaches using the new proposed equation for predicting the SLC of the simulated walls with openings are presented in Table 29 and Figure 60. As before, the accuracy was evaluated in terms of the minimum, maximum, average, standard deviation, coefficient of variation (CoV), 5th percentile, and 95th percentile of the ratio Vn,pred/Vmax,avg for each wall. The Mean Error (ME) and the Root Mean Squared Error (RMSE) were also calculated to evaluate the levels of precision of the approaches. These results do not consider any strength-reduction factors.
Figure 60: Results of the approaches for the shear capacity of the perforated walls.
(a) SA1 (b) SA2
(a) SA3 (b) SA4 Source: Author.
Avg 5th 95th
0.4 0.6 0.8 1.0 1.2 1.4 1.6
58 61 64 67 70 73 76 79 82 85 88 91 94 97 Vn,pred / Vmax,avg
Wall Number
Avg 5th 95th
0.4 0.6 0.8 1.0 1.2 1.4 1.6
58 61 64 67 70 73 76 79 82 85 88 91 94 97 Vn,pred / Vmax,avg
Wall Number
Avg 5th 95th
0.4 0.6 0.8 1.0 1.2 1.4 1.6
58 61 64 67 70 73 76 79 82 85 88 91 94 97 Vn,pred / Vmax,avg
Wall Number
Avg 5th 95th
0.4 0.6 0.8 1.0 1.2 1.4 1.6
58 61 64 67 70 73 76 79 82 85 88 91 94 97 Vn,pred / Vmax,avg
Wall Number
Table 29: Statistical comparison between the approaches for the shear capacity of the perforated walls.
Approach Vn,pred/Vmax,avg ME
(kN)
RMSE (kN) Min. Max. Avg. STDV CoV (%) 5th PCTL 95th PCTL
SA1 0.86 1.23 1.03 0.10 9.7 0.89 1.20 -15.2 48.9
SA2 0.59 0.94 0.80 0.10 12.1 0.62 0.92 91.2 100.4
SA3 0.64 0.97 0.86 0.07 7.8 0.73 0.94 66.5 73.2
SA4 0.87 1.15 0.99 0.06 6.2 0.89 1.10 4.8 29.8
Source: Author.
SA1 estimated the SLC of the perforated walls with an average ratio Vn,pred/Vmax,avg of 1.03 (CoV = 9.7%) and with the 5th and 95th percentiles equal to 0.89 and 1.20, respectively, which were close to 1.0, but with a slight tendency for overestimation, confirmed by a negative value of the ME (-15.2 kN). The main deficiency of this approach is to consider only the reduction of the effective horizontal cross-sectional area and not to consider the opening height.
The results show that SA2 made the worst predictions among the four approaches. The average and maximum ratio of Vn,pred/Vmax,avg of 0.80 and 0.94 indicate that all walls had their shear capacity underpredicted. This underestimation reached up to 41%, being larger for walls with two openings in the same story (walls 61, 66, 67, 77, 80, 81, 88, 91, and 92). The ME and the RMSE of 91.2 kN and 100.4 kN, respectively, were the highest among all approaches.
SA3 also underpredicted the shear capacity of all walls, but the values of the ratio Vn,pred/Vmax,avg were closer to 1.0 and the variation and errors were smaller compared to SA2.
The average and the maximum ratio of Vn,pred/Vmax,avg were 0.86 and 0.97, respectively, with a CoV of 7.8%. This approach also did not predict accurately the shear capacity of walls 66, 80, and 91, which had two door openings in the same story. The inaccuracy in predicting the shear capacity of walls with two openings with SA2 and SA3 may be related to the simplified assumption that the load carried by each pier and the strength of each pier are identical.
SA4 has the best statistical indicators among the four approaches assessed for the prediction of the SLC of perforated walls. The average ratio of Vn,pred/Vmax,avg was equal to 0.99 with a CoV of only 6.2% and with values of the 5th and 95th percentiles varying only approximately ±10%. The ME of 4.8 kN and the RMSE of 29.8 kN for SA4 were the smallest errors among all approaches. SA4 also predicted the SLC of the walls with two openings in the same story accurately, even when the openings were different (walls 67, 81, and 92). It can be noted that the predictions passed from a condition of underestimation (walls 59 to 74) to a condition of overestimation (walls 75 to 96) as the number of stories was increased. This might be explained by the fact that this approach is based on the strength of the piers and does not account for the global aspect ratio of the wall.