Table 6 gives the oscillation period of the structure for the ten primary modes in both SSI and fixed support cases. Regarding Table 6, the mass participation factors differ in the same modes with the effects of SSI. To verify the results, the relation of the fundamental period according to the NTC regulation, which is described as T1 = 0.0187h, has been used. By placing the height of the structure, h, which is equal to 5.12 m, in the mentioned equation; 0.096 sec is obtained, which is very close to the fixed support first period from numerical analysis, which is 0.093 sec. In order to perform a preliminary study on the effect of finite element mesh size on response accuracy, frequency analysis was carried out on a large number of fixed support models with different element sizes. Figure 14 illustrates a sensitivity analysis concerning the first three vibration periods and the element number. The investigation shows that convergence is achieved once the number of finite elements is increased.
Table 6
Characteristics of the first ten vibration modes.
Mode
|
Base condition
|
Period (s)
|
Participation factors
|
X-Trans
|
Z-Trans
|
X-Rot
|
Y-Rot
|
Z-Rot
|
1
|
Fixed support
|
0.093
|
0.00
|
1.59
|
3.79
|
0.00
|
0.00
|
SSI
|
0.216
|
0.00
|
1.64
|
15.50
|
0.00
|
0.00
|
2
|
Fixed support
|
0.063
|
0.00
|
0.00
|
0.01
|
0.02
|
0.01
|
SSI
|
0.201
|
0.00
|
0.00
|
0.00
|
47.59
|
0.00
|
3
|
Fixed support
|
0.059
|
0.23
|
0.00
|
0.00
|
7.61
|
0.72
|
SSI
|
0.176
|
1.56
|
0.00
|
0.00
|
0.00
|
10.52
|
4
|
Fixed support
|
0.058
|
1.06
|
0.00
|
0.01
|
1.46
|
3.38
|
SSI
|
0.162
|
0.92
|
0.00
|
0.00
|
0.00
|
1.23
|
5
|
Fixed support
|
0.054
|
1.63
|
0.00
|
0.00
|
0.02
|
4.05
|
SSI
|
0.158
|
0.00
|
0.00
|
0.00
|
0.00
|
0.00
|
6
|
Fixed support
|
0.047
|
0.00
|
0.00
|
0.02
|
1.49
|
0.00
|
SSI
|
0.157
|
0.00
|
0.00
|
0.00
|
13.65
|
0.00
|
7
|
Fixed support
|
0.045
|
0.00
|
0.12
|
0.67
|
0.02
|
0.01
|
SSI
|
0.152
|
0.00
|
0.04
|
0.87
|
0.00
|
0.00
|
8
|
Fixed support
|
0.041
|
0.00
|
0.00
|
0.06
|
0.36
|
0.04
|
SSI
|
0.150
|
0.07
|
0.00
|
0.00
|
0.00
|
1.07
|
9
|
Fixed support
|
0.039
|
0.00
|
0.00
|
0.01
|
1.47
|
0.03
|
SSI
|
0.149
|
0.00
|
0.22
|
11.93
|
0.00
|
0.00
|
10
|
Fixed support
|
0.039
|
0.02
|
0.00
|
0.96
|
0.23
|
0.10
|
SSI
|
0.146
|
0.00
|
0.06
|
39.51
|
0.00
|
0.00
|
Figure 15 shows the first five shape modes for the given finite element models. It can be seen that the important modes in terms of displacement and rotation considering the SSI are different with the fixed support case. Thus, it can be concluded that the SSI affects the shape of the modes, so that by comparing the first 60 modes of both models, only two common modes were found. Table 7 shows a comparison between common modes. Accordingly, the vibration period of the modes related to the model considering SSI is much longer than the fixed support model. Period ratio is usually provided by seismic codes to evaluate the response of soil-structure systems. According to Table 7, the period ratio ranges from 2.32 to 3.26 for the first two common modes. On the other hand, the ASCE 7 standard [63] provides the period ratio as follows:
where \(\tilde {T}\)= period of the SSI model, T = fixed support period, \(K_{{{\text{fixed}}}}^{*}\)= stiffness of the fixed support model, h = structure height, kx = foundation transition stiffness and kθ = foundation rotation stiffness. Transitional and rotational stiffness of the foundation were calculated according to the geotechnical report in Table 3. For this purpose, the equivalent modulus of elasticity was estimated as 34.32 MPa by using the weight averaging of the elasticity modulus of various soil layers. The dimensions of foundation have been taken 9 m × 14 m. Finally, the ratio of the period of the SSI model to the fixed support model equals 3.27. It is observed that the period ratio derived by the ASCE 7–16 is in good consonant with the results of numerical analysis.
A comparison between the capacity curves of the structure was performed in both situations, considering the SSI and the fixed support models. Figure 16 shows the capacity curves from the pushover analysis. On each response curve, three different points A, B, and C are marked; structure yielding occurrence (A); appearance of maximum tensile stress at the heel of the masonry wall, εtu = 0.15%, which is accompanied through an obvious alter in the diagram slope (B), and the peak displacement value (C). As shown in Fig. 16, the SSI effects produce a reduction of base shear and initial stiffness, and increases the displacement capacity. The effects of SSI in the X-direction reduces 68.7% and 69.9% of the base shear at the yield and collapse states, respectively, as well as increases 3.66 and 5.17 times the displacement at the yield and collapse points. In addition, for direction Z, SSI reduces 71.7% and 63.8% of the base shear at the yield and collapse states, respectively, as well as increases 1.49 and 2.11 times the corresponding displacements. As expected, the soil compliance has a higher impact within the X-direction due to the greater stiffness provided by the piers’ presence.
Table 7
Vibration periods of common modes.
Mode
|
Base condition
|
Period (s)
|
Period ratio
|
1
|
Fixed support
|
0.093
|
2.32
|
1
|
SSI
|
0.216
|
5
|
Fixed support
|
0.054
|
3.26
|
3
|
SSI
|
0.176
|
Figure 17 shows the progress of damage by increasing load in three stages corresponding to points A, B, and C on the pushover curves. According to Fig. 17, the damage onset is accompanied by tension cracks in the footwall joint. Then, in the X-direction, by creating cracks on the right and left sides of the short arch and dome, the dome and the short arch collapse and finally the whole structure are destroyed by the overturning of the pier walls. In the Z-direction, with the growth of cracks in the area of the bearing walls-long arch-short arc-dome and their connection to each other, as well as the appearing the cracks in the connection of the short arc to the wall, the entire roof of the Bazaar collapsed and the entire structure is destroyed by the overturning of the masonry walls. In the fixed support model, it is observed that the damage is more distributed in the entire structure because of the rigid base condition. The results of analyses indicate that plastic points and damage distribution are fundamentally modified by the SSI effects. As a consequence, the flexible foundation affects substantially the tension strain arrangement and afterwards the local failure mechanisms.
Figure 18 compares the capacity spectrum and the demand spectrum within the ADRS style. As shown, in design-basis earthquake demand, the fixed support model continues linearly in the X-direction whereas it hardly withstands against collapsing in the Z-direction. However, the SSI model collapses completely under the DBE demand. Moreover, the structure does not have the required capacity for MCE demand at all. Comparing the results of the fixed support model with the SSI case shows that the soil deformability leads to increment displacement, reduce base shear and factor of safety. Thus, the above-mentioned numerical simulations illustrate that the SSI effects are damaging to the structure seismic behavior, and the SSI omission overestimates the structure capacity and induces irrational responses.
For the given ground motion records, Fig. 19 shows the distribution of tensile damage in two models, fixed support and SSI cases. According to this Figure, the state of the structure in the fixed support model is better than the SSI model. Due to the Duzce and Tottori records, the structure maintains its stability and it is destroyed under the Manjil record. It is observed that the state of the structure in the SSI model is critical, so that the structure goes under complete collapse under all ground motion records. This time history results can be considered quietly match with the results of N2 method, which confirm each other. Examining the form of damage, it can be concluded that the destruction progress of the structure under all three records is as follows: The collapse of the dome with a short arc and then the collapse of the long arc, and finally the overthrow of the upright walls. The validity of these results is increased by comparing them with the pushover results in Fig. 20. As shown in Fig. 20, the damage distribution has many similarities in time history analysis and pushover procedure. So that, damage in time history analysis is a collection of damage in both directions X and Z. Since the structure is subjected to earthquake record components in both directions, this result is rational. Generally, it can be expressed that the structure collapse is not only due to the growth of cracks in each area but also due to the joining of the cracks, which causes the failure of the whole structure. As mentioned before, this is due to the fact that the continuous and interconnected performance in the force transition system has a key part for the Bazaar structure stability.
The base shear using time history analysis for the selected seismic record set is shown in Fig. 21. According to this Figure, the foundation flexibility reduces the base shear compared to the corresponding fixed support model. The value of this decrease for the Duzce record in the directions X and Z is respectively, -4.4% and 44.8%; for the Manjil record in the directions X and Z is respectively, 48.8% and 65.2%; and for the Tottori record in the directions X and Z is respectively, 79.5% and 67.0%. The maximum base shear from the pushover analysis for each direction is also shown as a horizontal line in the diagrams. It is observed that the maximum base shear time history is less than the maximum base shear from the pushover analyses, except in SSI model where, depending on the ground motion record, the base shear time history may go more than the maximum base shear in the pushover analysis (Fig. 21a).
Figure 22 compares the displacement time histories due to the considered ground motion records. According to this Figure, in the fixed support model, the displacement time histories are less than the pushover procedure. For the SSI model, the displacement values have increased to such an extent that in some places it reached the displacements from the pushover analysis. In the case of a fixed support, the displacement of the X-axis is much less than the Z-axis; however, in the model considering SSI, this difference is not observed. It is noted that such pattern was also observed in the pushover capacity curves as discussed earlier. In addition, the displacement value in the SSI model has increased compared to the fixed support model. This increase for the Duzce ground motion record in the directions X and Z is respectively, 9.43 and 5.28 times. For the Manjil record in the directions X and Z is respectively, 8.19 and 6.57 times. For the Tottori record in the directions X and Z is respectively, 3.87 and 3.05 times.