LIST OF SYMBOLS
5. SIMPLIFIED FRAME MODELS
5.3 NON-LINEAR BRACED FRAME MODEL .1 Modeling development
5.3.2 Assessment of model performance
The model performance was evaluated by comparing the initial lateral stiffness, lateral load capacity and corresponding lateral displacement at the top of the walls against the data of the experimental tests. The FE models developed in the previous chapters were also used in the comparisons, mainly as a reference for the situation with the higher pre-compression (𝜎 = 0.2𝑓𝑚′), as done in section 5.2.3. The envelope curves of walls with low and high axial loads are shown in Figure 121 and Figure 122, respectively, while a summary of results is presented in Table 39. Model BF-A represents the original response from the modeling, and model BF-B has the response of model BF-A with an imposed post-peak of 0.8Vmax limited to the ultimate drifts of 0.4% and 0.2% for the cases with low and high pre-compression, respectively.
Furthermore, the lateral deflected shapes of the experimental and numerical walls with low and high axial pre-compression are plotted in Figure 123 and Figure 124 for some levels of the maximum lateral load: 0.2Vmax, 0.4Vmax, 0.6Vmax, 0.8Vmax, Vmax, and 0.8Vmax*, this last one corresponding to the instant when the load had dropped 20% after the peak.
As for the linear frame models, the distribution of loads throughout the wall elements in the first story of the non-linear model BF is compared with results from the FE model. The results are presented in Table 40, taking walls D1-2 as an example and considering different lateral load levels.
Figure 121: Envelope load-displacement curves of the experimental tests and models with the lower pre-compression (𝜎 = 0.04𝑓𝑚′).
(a) Walls W1 and W2 (b) Walls D1 and D2 Source: Author.
Figure 122: Envelope load-displacement curves of the FE model and models BF with the higher pre-compression (𝜎 = 0.2𝑓𝑚′).
(a) Walls W1 and W2 (b) Walls D1 and D2 Source: Author.
Table 39: Results of the experimental tests and models with the low and high pre-compressions.
Wall
(axial load case) Response
Stiffness Force Top displacement K0
(kN/mm) ΔK0
(%)
Vmax
(kN) ΔVmax
(%)
dVmax
(mm) ΔdVmax
(%) du
(mm) Δdu
(%)
W1 & W2 (𝜎 = 0.04𝑓𝑚′)
Model BF 37.0 --- 101.9 --- 13.3 --- 17.0* ---
FE Model 72.9 -49.2 98.9 3.1 11.7 13.4 16.4 3.9
Exp. Wall W1 82.8 -55.3 97.2 4.9 11.8 12.5 15.0 13.6
Exp. Wall W2 48.0 -22.9 104.1 -2.1 11.7 13.4 16.3 4.5
Avg. Experimental 63.7 -41.9 100.7 1.3 11.8 12.5 15.7 8.5
0.00 0.12 0.24 0.36 0.48 0.60
0 20 40 60 80 100 120
0 5 10 15 20 25
Top drift (%)
Lateral Load (kN)
Lateral Displacement (mm) Model BF-A Model BF-B FE Model Exp. Wall W1 Exp. Wall W2 Avg. Experimental
0.00 0.12 0.24 0.36 0.48 0.60
0 20 40 60 80 100 120
0 5 10 15 20 25
Top drift (%)
Lateral Load (kN)
Lateral Displacement (mm) Model BF-A Model BF-B FE Model Exp. Wall D1 Exp. Wall D2 Avg. Experimental
0.00 0.12 0.24 0.36 0.48 0.60
0 30 60 90 120 150 180
0 5 10 15 20 25
Top drift (%)
Lateral Load (kN)
Lateral Displacement (mm) Model BF-A Model BF-B FE Model
0.00 0.12 0.24 0.36 0.48 0.60
0 30 60 90 120 150 180
0 5 10 15 20 25
Top drift (%)
Lateral Load (kN)
Lateral Displacement (mm) Model BF-A Model BF-B FE Model
D1 & D2 (𝜎 = 0.04𝑓𝑚′)
Model BF 36.0 --- 99.5 --- 13.5 --- 17.0* ---
FE Model 61.9 -41.8 98.1 1.4 13.9 -3.2 19.0 -10.5
Exp. Wall D1 52.6 -31.6 96.8 2.8 15.5 -13.2 20.0 -15.0 Exp. Wall D2 46.7 -22.9 102.0 -2.5 17.0 -20.9 22.3 -23.7 Avg. Experimental 49.5 -27.3 99.4 0.1 16.2 -17.0 21.5 -20.9 W1 & W2
(𝜎 = 0.2𝑓𝑚′)
Model BF 37.0 --- 130.2 --- 10.3 --- 8.6* ---
FE Model 78.3 -52.7 155.3 -16.2 6.0 72.2 9.0 -5.3
D1 & D2 (𝜎 = 0.2𝑓𝑚′)
Model BF 36.0 --- 116.7 --- 11.9 --- 8.6* ---
FE Model 69.3 -48.1 146.7 -20.5 5.7 108.4 10.0 -14.3
*Value corresponding to the imposed ultimate drifts in Model BF-B.
Source: Author.
Figure 123: Experimental and numerical deflected shapes of walls with the lower pre-compression (𝜎 = 0.04𝑓𝑚′) for different lateral load levels.
(a) Walls W1 and W2 (b) Walls D1 and D2 Source: Author.
Figure 124: Deflected shapes of the FE model and models BF with the higher pre-compression (𝜎 = 0.2𝑓𝑚′) for different lateral load levels.
(a) Walls W1 and W2 (b) Walls D1 and D2 Source: Author.
1st Slab 2nd Slab 3rd Slab
0 715 1430 2145 2860 3575 4290
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0
Wall height (mm)
Lateral displacement (mm)
0.2Vmax 0.4Vmax
0.6Vmax 0.8Vmax
Vmax 0.8Vmax*
Experimental - - - -Model BF
1st Slab 2nd Slab 3rd Slab
0 715 1430 2145 2860 3575 4290
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0
Wall height (mm)
Lateral displacement (mm)
0.2Vmax 0.4Vmax
0.6Vmax 0.8Vmax
Vmax 0.8Vmax*
Experimental - - - -Model BF
1st Slab 2nd Slab 3rd Slab
0 715 1430 2145 2860 3575 4290
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0
Wall height (mm)
Lateral displacement (mm)
0.2Vmax 0.4Vmax
0.6Vmax 0.8Vmax
Vmax 0.8Vmax*
FE Model - - - -Model BF
1st Slab 2nd Slab 3rd Slab
0 715 1430 2145 2860 3575 4290
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0
Wall height (mm)
Lateral displacement (mm)
0.2Vmax 0.4Vmax
0.6Vmax 0.8Vmax
Vmax 0.8Vmax*
FE Model - - - -Model BF
Table 40: Element loads in the first story of walls D1-2 for the non-linear FE and BF models.
Lateral load level
(kN)
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) 15 FE Model -20.9 7.3 14.0 -60.1 9.2 15.8 -0.3 5.1 1.2
Model BF -22.4 8.0 16.2 -58.5 8.5 16.6 -0.3 4.8 2.7 50 FE Model 18.4 13.3 17.5 -99.5 36.6 72.9 -1.3 15.8 3.5 Model BF 19.6 25.9 48.6 -100.5 24.5 48.9 -0.2 19.3 11.0 90 FE Model 64.7 28.0 37.7 -145.8 62.1 128.4 5.2 25.6 7.2
Model BF 112.5 45.6 35.2 -193.2 45.2 35.5 -0.1 56.3 31.9 99 FE Model 75.7 29.5 41.4 -156.4 68.4 138.6 5.9 28.1 8.2
Model BF 149.5 50.2 15.1 -230.4 49.3 15.3 -0.1 68.6 38.8 Source: Author.
As can be seen in Figure 121, there was good agreement between the envelope curves of the model BF and the experimental walls up to the peak load. Unlike the FE model, model BF-A was not able to simulate the post-peak behavior of the walls, not presenting the expected strength degradation. This problem was dealt with in model BF-B by imposing an ultimate load of 0.8Vmax with a corresponding ultimate displacement limited to a drift of 0.4%.
The average experimental lateral stiffness, maximum lateral load, and corresponding lateral displacement at the top of walls W1-2 were 63.7 kN/mm, 100.7 kN, and 11.8 mm, respectively. The model BF for these walls resulted in a lateral stiffness of 37 kN/mm, a peak load of 101.9 kN, and a corresponding displacement of 13.3 mm, which were 41.9% lower, 1.3% higher, and 12.5% higher, respectively, than the experimental results. For walls D1-2, the average experimental results for the lateral stiffness, peak load, and the corresponding displacement were 49.5 kN/mm, 99.4 kN, and 16.2 mm, respectively, against 36 kN/mm (- 27.3%), 99.5 kN (+0.1%) and 13.5 mm (-17%) obtained from the model BF. With the imposed ultimate drift of 0.4% to the model BF-B, the ultimate displacement was 8.5% higher for walls W1-2 and 20.9% lower for walls D1-2.
Regarding the walls with the higher pre-compression, substantial differences exist between the envelope curves of model BF and the FE model, even in the pre-peak stage, Figure 122. The maximum lateral load differences were reasonable, with the model BF being more conservative than the FE model by 16.2% for walls W1-2 and 20.5% for walls D1-2. However, model BF was up to 52.7% more flexible than the FE model, which directly influenced a displacement at the peak load up to 108.4% higher. A possible reason for this contrast may be the absence of the interaction between axial load and moment in the strength capacity of the flexural hinges; many attempts were made to include this interaction, but all stopped due to
numerical convergence problems. Dealing with this limitation, imposing the drift limit of 0.2%, as in model BF-B, resulted in acceptable conservatism.
The deflected shapes of the walls with the lower pre-compression obtained with model BF are compatible with the experimental walls up to the peak load, as observed in Figure 123.
The major difference between inter-story drifts was found in the third story at the stage of maximum lateral load; the values for the Model BF were approximately 29% higher and lower than the experimental walls W1-2 and D1-2, respectively. Concerning the walls with the higher axial load, the comparison of the deflected shapes shown in Figure 124 corroborates that model BF is significantly more flexible than the FE Model for all lateral load levels.
The results in Table 40 show that the reaction loads calculated at the bottom center point of each pier from model BF match the reaction loads integrated at the bottom center point of each pier in the linear phase of the FE model. As the lateral load increases and the wall behavior becomes more non-linear, significant differences in the axial force and moment are noted.
Unlike the FE model, Model BF presented similar values for the moment in both left and right wall piers, appearing to be incapable of representing this phenomenon correctly. Numerous variables were adjusted in an attempt to understand what causes this behavior, but nothing became apparent.