LIST OF SYMBOLS
5. SIMPLIFIED FRAME MODELS
5.2 LINEAR FRAME MODELS .1 Modeling development
5.2.2 Assessment of initial lateral stiffness, deflection, and load distribution
The response of the linear models is plotted together with the backbone curve of the experimental walls in Figure 109 to provide a graphical comparison. An overall view of the load-displacement curves is presented in Figure 109(a) and Figure 109(c), and a zoom of the initial stage is shown in Figure 109(b) and Figure 109(d), where it is possible to identify better the difference between the lateral stiffness of the curves. In these graphs, the lateral displacement is related to the wall top, and the lateral load is the total reaction load. Also, the elastic lateral deflected shapes of the experimental and numerical walls corresponding to the load point of approximately 15 kN are plotted in Figure 110. In addition to the curves, the results are presented in detail in Table 35, including the lateral displacement, and the inter-story and top drifts and stiffnesses of the experimental and numerical walls. The experimental results are an average envelope of the pull and push regimes, and the initial stiffness was defined as the tangent stiffness for a load of approximately 15 kN.
Figure 109: Top load-displacement curves of all linear models and experimental walls.
(a) Walls W1 and W2 – Overall view (b) Walls W1 and W2 – Zoom
(c) Walls D1 and D2 – Overall view (d) Walls D1 and D2 – Zoom Source: Author.
0 20 40 60 80 100 120
0 5 10 15 20 25
Lateral load (kN)
Lateral displacement (mm)
Experimental Exp. Linear
Model PF1 Model PF2
Model PF3 Model PF4
Model PF5 Model BF
0 10 20 30 40
0.0 0.5 1.0 1.5 2.0 2.5
Lateral load (kN)
Lateral displacement (mm)
Experimental Exp. Linear Model PF1 Model PF2 Model PF3 Model PF4 Model PF5 Model BF
0 20 40 60 80 100 120
0 5 10 15 20 25
Lateral load (kN)
Lateral displacement (mm)
Experimental Exp. Linear
Model PF1 Model PF2
Model PF3 Model PF4
Model PF5 Model BF
0 10 20 30 40
0.0 0.5 1.0 1.5 2.0 2.5
Lateral load (kN)
Lateral displacement (mm)
Experimental Exp. Linear Model PF1 Model PF2 Model PF3 Model PF4 Model PF5 Model BF
Figure 110: Experimental and numerical elastic deflected shapes of walls.
(a) Walls W1 and W2 (b) Walls D1 and D2 Source: Author.
Table 35: Experimental and numerical elastic results.
Wall Story Parameter Avg.
Test
Model PF1
Model PF2
Model PF3
Model PF4
Model PF5
Model BF
W1
&
W2
1st
δ (mm) 0.08 0.18 0.12 0.11 0.09 0.06 0.10
Drift (%) 0.005 0.013 0.008 0.007 0.007 0.004 0.007 K0 (kN/mm) 194.0 83.2 127.3 141.1 158.0 237.3 151.7 2nd
δ (mm) 0.15 0.55 0.31 0.27 0.24 0.15 0.25
Drift (%) 0.005 0.026 0.014 0.012 0.010 0.006 0.011 K0 (kN/mm) 193.4 41.1 77.3 91.2 106.6 175.2 99.4 3rd
δ (mm) 0.24 1.00 0.53 0.44 0.39 0.24 0.41
Drift (%) 0.006 0.032 0.015 0.012 0.011 0.007 0.011 K0 (kN/mm) 163.3 32.8 70.3 87.0 97.8 156.6 96.3 Top
δ (mm) 0.24 1.00 0.53 0.44 0.39 0.24 0.41
Drift (%) 0.006 0.023 0.012 0.010 0.009 0.006 0.009 K0 (kN/mm) 63.7 15.0 28.6 33.8 38.6 61.3 37.0
D1
&
D2
1st
δ (mm) 0.08 0.18 0.13 0.10 0.10 0.08 0.10
Drift (%) 0.006 0.013 0.009 0.007 0.007 0.006 0.007 K0 (kN/mm) 180.6 83.2 116.8 144.1 156.0 181.9 145.3 2nd
δ (mm) 0.17 0.55 0.35 0.26 0.24 0.21 0.26
Drift (%) 0.006 0.026 0.015 0.011 0.010 0.009 0.011 K0 (kN/mm) 180.8 41.1 68.8 94.5 105.6 122.4 93.9 3rd
δ (mm) 0.30 1.00 0.58 0.43 0.39 0.34 0.42
Drift (%) 0.010 0.032 0.016 0.012 0.011 0.009 0.011 K0 (kN/mm) 109.4 32.8 64.1 90.2 97.4 112.4 97.6 Top
δ (mm) 0.30 1.00 0.58 0.43 0.39 0.34 0.42
Drift (%) 0.007 0.023 0.014 0.010 0.009 0.008 0.010 K0 (kN/mm) 49.5 15.0 25.8 35.0 38.2 44.3 36.0
Source: Author.
1st Slab 2nd Slab 3rd Slab
0 715 1430 2145 2860 3575 4290
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Wall height (mm)
Lateral displacement (mm)
Model PF1 Model PF2 Model PF3 Model PF4
Model PF5 Model BF
Avg. Exp.
1st Slab 2nd Slab 3rd Slab
0 715 1430 2145 2860 3575 4290
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Wall height (mm)
Lateral displacement (mm)
Model PF1 Model PF2 Model PF3 Model PF4
Model PF5 Model BF
Avg. Exp.
Examination of Figure 109, Figure 110, and the results in Table 35 reveals that Model PF1 was the most flexible, being up to 82.4% and 76.5% less stiff than the experimental walls when compared the inter-story and top lateral stiffnesses, respectively. Regarding the inter- story and top lateral drifts, Model PF1 achieved, respectively, values up to six and four times higher than the experimental walls. This implies that using a cantilever model to simulate coupled perforated walls is highly conservative. Model PF2 was almost twice as stiff as Model PF1 but still notably different to the tested walls.
The stiffness became greater and, consequently, the deflections decreased as rigid offsets were applied in the portal frame modeling (Models PF3, PF4, and PF5). Including rigid offsets on the horizontal and vertical elements (Model PF5) resulted in lateral deflections and stiffnesses closer to those of the experimental walls. The differences were about ±4% and ±10%
related to the top of walls W1-2 and D1-2, respectively, and ±20% and ±3% related to the inter- story values for walls W1-2 and D1-2, respectively. Exceptionally, the inter-story drift and stiffness of the second story in walls D1-2 deviated from this small range, being in Model PF5 up to 48% more conservative than the experimental tests. Comparing Model PF5 (horizontal and vertical rigid offsets) with Models PF3 and PF4 (only horizontal rigid offsets), it may be seen that vertical offsets were more crucial for the lateral deflection and stiffness in walls W1- 2 (with window openings) than in walls D1-2 (with door openings). This distinction can be credited to the masonry regions below the openings in walls W1-2 that did not exist in walls D1-2. Kappos et al. (2002) also observed minimal differences using a similar approach with horizontal and vertical rigid offsets.
In general, the experimental walls had elastic inter-story drifts similar in the first and second stories, but higher in the third story, mainly for walls D1-2, while the frame models presented significant differences in the inter-story drifts from the first to the second story and minor changes from the second to the third story. The portal frame models had inter-story drifts and stiffnesses with more remarkable differences from the tests for the third story in walls W1-2 and for the second story in walls D1-2. The smallest differences were observed for the first story in both types of walls. In turn, the braced frame model had more notable variations from the tests for the second story in both types of wall, and the lowest deviations for the first story in walls W1-2 and for the third story in walls D1-2.
Model BF presented deflections and lateral stiffness between Models PF3 and PF4, with conservative differences up to 70% to 30% compared to the experimental walls W1-2 and D1-2, respectively, related to the wall top and up to 95% related to the inter-story values.
The distribution of loads throughout the wall elements in the first story (the critical story) is analyzed by comparing the distribution with the results from the FE model developed in the previous chapters. The internal loads were extracted from the FE model using the software tool that integrates the stress in the elements in the same alignment; e.g., all elements of the first course of the right wall pier, as shown in Figure 111. To compare the BF-results, in which each pier has two vertical elements, the reaction loads at the bottom of both vertical elements were transferred to the bottom center of their respective pier, as illustrated in Figure 112. The results are presented in Table 36, taking walls D1-2 as an example and considering a lateral load level of approximately 15 kN.
Figure 111: Internal loads extraction in the FE model.
Source: Author.
Figure 112: Internal loads extraction in the model BF; dimensions in mm.
Source: Author.
The comparison of internal loads between the portal frame models and the FE model in the linear phase confirms the conservatism of models PF1 and PF2 and better approximations as vertical rather than only horizontal rigid offsets are implemented in the elements. As verified in Table 36, Models PF3-5 and BF could represent the load distribution with less discrepancy.
Table 36: Internal loads of elements in the first story of walls D1-2.
Model
Left Pier Right Pier Beam
N (kN)
V (kN)
M (kN·m)
N (kN)
V (kN)
M (kN·m)
N (kN)
V (kN)
M (kN·m)
FE Model -20.9 7.3 14.0 -60.1 9.2 15.8 -0.3 5.1 1.2
Model PF1 -40.4 7.5 32.1 -40.4 7.5 32.1 0.0 0.0 0.0
Model PF2 -30.5 6.5 17.4 -50.3 6.5 17.4 0.0 1.6 1.6
Model PF3 -20.7 8.7 16.9 -60.1 8.7 16.9 0.0 5.7 2.7
Model PF4 -19.3 8.5 14.3 -61.5 8.5 14.3 0.0 7.3 2.1
Model PF5 -19.9 8.5 15.1 -60.8 8.5 15.1 0.0 6.5 1.8
Model BF -22.4 8.0 16.2 -58.5 8.5 16.6 -0.3 4.8 2.7
Source: Author.