Luận án Nghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đất

n contrast, the maximum incremental normal forces in the full-slip case are smaller than that of the no-slip case (Figure 2.7b and Figure 2.7d). Figure 2.5. Deformed model and displacement contours in circular tunnel model for no-slip condition Figure 2.6. Deformed model and displacement contours in circular tunnel model for full-slip condition 37 Wang solution: a) Incremental Bending Moments b) Incremental Normal Forces Numerical solution (FDM): c) Incremental Bending Moments d) Incremental Normal Forces Figure 2.7. Distribution of the incremental internal forces in the circular tunnel by Flac3D and Wang solution. 2.2. Validation of circular tunnel under seismic loading In the section below, a parametric study is conducted to highlight the behavior of circular tunnel lining subjected to quasi-static loadings considering the effect of Young’s modulus Es, horizontal seismic acceleration aH, and tunnel lining thickness 45° No-slip case: Mmax = 0.738 MNm/m Full slip case: Mmax = 0.845 MNm/m 45° No-slip case: Nmax = 0.894 MN/m Full slip case: Nmax = 0.173 MN/m 45° No-slip case: Mmax = 0.741 MNm/m Full slip case: Mmax = 0.834 MNm/m 45° No-slip case: Nmax = 0.903 MN/m Full slip case: Nmax = 0.169 MN/m 38 t variations. Parameters of the soil and tunnel lining presented in Table 2.1 are adopted for the reference case study. Both maximum and minimum incremental bending moments are presented. They are named ‘extreme incremental bending moment’. Similarly, extreme incremental normal forces representing both the maximum and minimum incremental normal forces induced in the tunnel lining. 2.2.1. Effect of the peak horizontal seismic acceleration (aH) A parametric study was conducted to investigate the seismic loading magnitude effects represented here by the maximum horizontal acceleration, aH. The maximum horizontal acceleration is varied in the range between 0.05 and 0.75 g, corresponding to the respectively maximum shear strains, γmax, of 0.038 and 0.58%. The other parameters of the reference case in Table 2.1 are used. The following conclusions can be deduced from Figure 2.8: a) Extreme incremental bending moments b) Extreme incremental normal forces Figure 2.8. Effect of aH on the extreme incremental internal forces of the circular tunnel lining -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 E xt re m e In cr em en ta l B en di ng M om en t M (M N m /m ) aH (g) Mmax_FDM_ns Mmax_FDM_fs Mmax_Wang_ns Mmax_Wang_fs Mmin_FDM_ns Mmin_FDM_fs Mmin_Wang_ns Mmin_Wang_fs -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 E xt re m e In cr em en ta l N or m al F or ce s N (M N /m ) aH (g) Nmax_FDM_ns Nmax_FDM_fs Nmax_Wang_ns Nmax_Wang_fs Nmin_FDM_ns Nmin_FDM_fs Nmin_Wang_ns Nmin_Wang_fs 39 - For both the no-slip and full-slip conditions, numerical results show a very good agreement with the analytical solution. A discrepancy of approximately 1% for both the extreme incremental bending moments and normal forces is obtained, - The absolute values of the extreme incremental bending moments and normal forces increased gradually with the aH increase from 0.05 g to 0.75 g. Incremental bending moments for both no-slip and full slip conditions are strongly dependent on the aH value (Figure 2.8a). However, while incremental normal forces in the tunnel lining for the no-slip condition are strongly affected by the aH value, insignificant incremental normal forces variations due to aH for the full slip condition are observed (Figure 2.8b). 2.2.2. Effect of the soil Young’s modulus, Es The soil Young’s modulus is assumed to fall in a range from 10 to 350 MPa. The other parameters presented in Table 2.1, based on the reference case study were used as the input data. The numerical results obtained by using the Flac3D comparison with the analytical Wang’s method for the full slip and no-slip conditions are presented in Figure 2.9. The following comments can be deduced: - Figure 2.9 shows a very good agreement of the incremental bending moments and normal forces induced in the tunnel lining under seismic loading, obtained by the numerical model and the analytical solution for both no-slip and full slip conditions when considering the Es variation. The maximum difference is smaller than 2 %. - The extreme incremental bending moments are strongly dependent on the Es value as seen in Figure 2.9a. The absolute values of the extreme incremental bending moments were reached for Es values close to 50 MPa. There was a rapid decrease of the absolute extreme incremental bending moments when the Es value reduces from 25 to 10 MPa. This can be explained by the fact that when the tunnel lining is stiffer than the ground, the tunnel lining tends to resist the ground displacements. When the Es values are larger than 50 MPa, the tunnel structure is more flexible than the ground. As a consequence, the tunnel lining will amplify the distortion compared with the soil 40 shear distortions in the free field. An increase of Es induces a decrease of the absolute extreme incremental bending moments. This correlation of the extreme incremental bending moments is observed in both full slip and no-slip conditions. It should be noted that for the same Es value, the absolute extreme incremental bending moments induced in the tunnel lining for the no-slip condition are always 10% to 15% smaller than the full slip ones. - While the extreme incremental normal forces in the full slip conditions are insignificantly dependent on the Es value (Figure 2.9b), an increase of Es can cause a rapid increase of the maximum and minimum incremental normal forces in the tunnel lining for the no-slip condition. As predicted, incremental normal forces for the no- slip condition are always larger than the full slip ones. a) Extreme incremental bending moments b) Extreme incremental normal Forces Figure 2.9. Effect of Es on the incremental internal forces of the circular tunnel lining 2.2.3. Effect of the lining thickness, t The tunnel lining thickness was assumed to vary between 0.2 to 0.8 m, corresponding to the common lining thickness vs tunnel dimension ratio of 3% to 8.5% [60], while the other parameters are based on the reference case assumed in -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 0 50 100 150 200 250 300 350 Ex tr em e In cr em en ta l B en di ng M om en t M (M N m /m ) Young's Modulus, Es (MPa) Mmax_FDM_ns Mmax_FDM_fs Mmax_Wang_ns Mmax_Wang_fs Mmin_FDM_ns Mmin_FDM_fs Mmin_Wang_ns Mmin_Wang_fs -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 50 100 150 200 250 300 350 Ex tr em e In cr em en ta l N or m al F or ce s N (M N /m ) Young's Modulus, Es (MPa) Nmax_FDM_ns Nmax_FDM_fs Nmax_Wang_ns Nmax_Wang_fs Nmin_FDM_ns Nmin_FDM_fs Nmin_Wang_ns Nmin_Wang_fs 41 Table 2.1. Similar to what happens when considering Young’s soil modulus Es and horizontal seismic acceleration aH effects, the results presented in Figure 2.10 show a very good agreement between analytical and numerical models for both no-slip and full slip conditions. The discrepancy is under 1% for both the incremental bending moments and normal forces. In general, the absolute extreme incremental bending moments and normal forces values gradually increase when the lining thickness t increases from 0.2 to 0.8 m. This concerns both the full slip and no-slip conditions. The incremental bending moments for the no-slip condition are always smaller than the full slip ones (Figure 2.10a). The biggest difference of 14% was obtained for a lining thickness of 0.8m. It should be noted that the incremental normal forces variations caused by the lining thickness increase are less significant when compared to the incremental bending moment ones (Figure 2.10a and Figure 2.10b). a) Extreme incremental bending moments b) Extreme incremental normal forces Figure 2.10. Effect of the lining thickness on the incremental internal forces in the circular tunnel lining Based on the above comparison between the analytical solution and numerical model when considering Young’s modulus Es, horizontal seismic acceleration aH, and tunnel lining thickness t, which show a very good agreement between the analytical -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 E xt re m e In cr em en ta l B en di ng M om en t M (M N m /m ) Lining thickness (m) Mmax_FDM_ns Mmax_FDM_fs Mmax_Wang_ns Mmax_Wang_fs Mmin_FDM_ns Mmin_FDM_fs Mmin_Wang_ns Mmin_Wang_fs -2 -1.5 -1 -0.5 0 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ex tr em e In cr em en ta l N or m al F or ce s N (M N /m ) Lining thickness (m) Nmax_FDM_ns Nmax_FDM_fs Nmax_Wang_ns Nmax_Wang_fs Nmin_FDM_ns Nmin_FDM_fs Nmin_Wang_ns Nmin_Wang_fs 42 solution and numerical model, it is reasonable to conclude that the circular tunnel numerical model developed can be used to investigate the behavior of circular tunnels subjected to seismic loadings. 2.3. Numerical simulation of the sub-rectangular tunnel under seismic loading Figure 2.11. Geometry and quasi-static loading conditions in the numerical model of a sub-rectangular tunnel In this section, a numerical model was developed for the sub-rectangular tunnels cased using similar soil parameters, lining material, and modeling processes to consider the static and seismic loadings introduced above. Only the tunnel shape is modified into a sub-rectangular geometry and the gravity effect is taken into consideration. The mesh consists of a single layer of zones in the y-direction, and the dimension of the elements increases as one moves away from the tunnel (Figure 2.11). The geometry parameters of sub-rectangular tunnels are presented in Figure 2.1 and other parameters presented in Table 2.1 are adopted. The numerical model is 120 m wide in the x-direction, and 40 m high in the z-direction and consists of approximately 5816 elements and 11870 nodes. The bottom of the model was blocked in all directions and the vertical sides were fixed in the horizontal one. 2.4. Parametric study of sub-rectangular tunnels in quasi-static conditions Deformed model and displacement vectors of sub-rectangular tunnel under seismic loading are shown in Figures 2.12 and 2.13. While Figure 2.14 introduces the incremental bending moments and normal forces induced in the sub-rectangular Pr es cr ib ed d is pl ac em en t γmax 43 tunnel linings subjected to seismic loadings and considering both no-slip and full slip conditions. Parameters of the reference case presented in Table 2.1 are adopted. Figure 2.12. Deformed model and displacement contours in Sub-rectangular tunnel model for no-slip condition Figure 2.13. Deformed model and displacement contours in Sub-rectangular tunnel model for full-slip condition 44 a) Incremental Bending Moment b) Incremental Normal Forces Figure 2.14. Distribution of the incremental bending moments and normal forces in the sub-rectangular tunnel Figures 2.14 and 2.7 allow having a clear understanding of the behavior of circular and sub-rectangular tunnel linings under seismic loadings. The positions at the tunnel periphery, where the extreme incremental internal forces are reached, are positioned. It can be seen from Figure 2.14 that extreme incremental bending moments and normal forces observed in the sub-rectangular tunnel appear at the tunnel lining corners where the smaller lining radii are located. In the following sections, a numerical investigation was conducted to highlight the behavior of a sub- rectangular tunnel compared with a circular shape. These two tunnels have the same utilization space area and are twice subjected to seismic loadings while considering the effect of parameters, like the horizontal seismic acceleration, soil deformation modulus, and lining thickness. Effects of the soil-lining interface condition are also investigated. 2.4.1. Effect of the peak horizontal seismic acceleration (aH) Shear strain values corresponding to a range of a maximum horizontal acceleration varying from 0.05g and 0.75g were adopted in this study. In general, high seismic loadings are implied by a high horizontal acceleration aH, and therefore No-slip case: Mmax = 0.900 MNm/m Full slip case: Mmax = 0.807 MNm/m 33° 33° No-slip case: Nmax = 0.791 MN/m Full slip case: Nmax = 0.159 MN/m 45 shear strain values of γmax, result in high absolute extreme incremental bending moments and normal forces. The relationship is quite linear (Figure 2.15). a) Incremental bending moments b) Incremental normal forces Figure 2.15. Effect of the aH value on the internal forces of circular and sub- rectangular tunnel linings The results presented in Figure 2.15a show that, for the no-slip condition, absolute extreme incremental bending moments in the sub-rectangular lining are 20% larger than the circular ones. Nevertheless, for the full slip condition, absolute extreme incremental bending moments in the circular lining are approximately 4% greater than the sub-rectangular ones. In the case of sub-rectangular linings, absolute extreme incremental bending moments for the full slip condition are always lower by about 10% than the no-slip ones. This relationship is opposite to the one observed in the cases of the circular-shaped tunnel (Figure 2.15a). It can be seen in Figure 2.15b that for both shapes of tunnels, the absolute extreme incremental normal forces for the no-slip condition are approximately 80% larger than the full slip ones. Unlike the incremental bending moments mentioned above, the absolute extreme incremental normal forces of the sub-rectangular lining are approximately 9% lower than the circular lining ones, for both the no-slip and full slip conditions and when changing the horizontal acceleration (Figure 2.15b). -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 E xt re m e In cr em en ta l B en di ng M om en t M (M N m /m ) aH (g) Mmax_SR_ns Mmax_SR_fs Mmax_Circular_ns Mmax_Circular_fs Mmin_SR_ns Mmin_SR_fs Mmin_Circular_ns Mmin_Circular_fs -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 E xt rm e In cr em en ta l N or m al F or ce s N (M N /m ) aH (g) Nmax_SR_ns Nmax_SR_fs Nmax_Circular_ns Nmax_Circular_fs Nmin_SR_ns Nmin_SR_fs Nmin_Circular_ns Nmin_Circular_fs 46 2.4.2. Effect of the soil’s Young’s modulus (Es) Soil Young’s modulus values are assumed to vary in the range from 10 to 350 MPa while keeping K0 equal to 0.5 and aH of 0.5g. The other parameters based on the reference case are assumed (Table 2.1). It can be seen from Figure 2.16 that: a) Incremental bending moments b) Incremental normal forces Figure 2.16. Effect of the Es value on the internal forces for the circular and sub- rectangular tunnel linings - For low Es values smaller than 50 MPa, an increase of Es induces an increase of the absolute extreme incremental bending moments. When the Es value is greater than 50 MPa, the increase of Es causes a decrease of the absolute extreme incremental bending moments (Figure 2.16a). It should be noted that the dependency of the absolute extreme incremental bending moments in the sub-rectangular tunnels on the Es value is insignificant compared to the circular-shaped tunnels (Figure 2.16a). It is also worth highlighting that, while the absolute extreme incremental bending moments of the circular tunnel for the no-slip condition are smaller than the corresponding full slip ones [47],[159], the absolute extreme incremental bending moments of the sub-rectangular tunnel for the no-slip condition are greater than the corresponding full slip ones. The behavior of sub-rectangular tunnels is different from -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 0 50 100 150 200 250 300 350 E xt re m e In cr em en ta l B en di ng M om en t M (M N m /m ) Young's Modulus, Es (MPa) Mmax_SR_ns Mmax_SR_fs Mmax_Circular_ns Mmax_Circular_fs Mmin_SR_ns Mmin_SR_fs Mmin_Circular_ns Mmin_Circular_fs -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 50 100 150 200 250 300 350 E xt re m e In cr em en ta l N or m al F or ce s N (M N /m ) Young's Modulus, Es (MPa) Nmax_SR_ns Nmax_SR_fs Nmax_Circular_ns Nmax_Circular_fs Nmin_SR_ns Nmin_SR_fs Nmin_Circular_ns Nmin_Circular_fs 47 circular-shaped tunnels. The same conclusion was also obtained when considering the horizontal seismic acceleration aH. - Figure 2.16a also shows greater absolute extreme incremental bending moments induced in sub-rectangular tunnels for the no-slip condition compared with circular tunnels having the same utilization space area. However, in the full slip condition, absolute extreme incremental bending moments in the circular tunnel are greater than the sub-rectangular ones for Es values smaller than approximately 150 MPa. When Es values are larger than 150 MPa, absolute extreme incremental bending moments developed in circular tunnels are smaller than in sub-rectangular tunnels. - Figure 2.16b indicates that an increase of Es value causes a significant corresponding increase of the absolute extreme normal forces in both sub-rectangular and circular tunnels for the no-slip condition. But it induces an insignificant change in absolute extreme incremental normal forces for the full slip condition. Absolute extreme incremental normal forces in the sub-rectangular tunnels are generally 9% smaller than for the circular ones. 2.4.3. Effect of the lining thickness (t) The lining thickness t is assumed to vary in the range between 0.2 to 0.8 m while keeping K0 value of 0.5, and aH value of 0.5g, and Es value of 100 MPa. Other parameters introduced in Table 2.1 were adopted. The results presented in Figure 2.17 indicate that the lining thickness has a great effect on the incremental internal forces for both sub-rectangular and circular tunnels and in both no-slip and full slip conditions. The relationship between the lining thickness and the incremental internal forces for the considered cases is quite linear. For the no-slip condition, absolute extreme incremental bending moments of the sub-rectangular linings are always larger than the circular ones (Figure 2.17a). The discrepancy declined gradually from 124% to 6%, corresponding to the lining thickness increase from 0.2 to 0.8 m. In the full slip conditions, considering a small lining thickness smaller than approximately 0.5m, the absolute extreme incremental 48 bending moments of the sub-rectangular linings are still larger than the circular ones, just like for the no-slip condition presented earlier. However, when the lining thickness is larger than 0. 5m, Figure 2.17a proves an opposite result. Thus, larger absolute extreme incremental bending moments on the circular tunnels are observed. It can be seen in Figure 2.17b that the incremental normal forces in the no-slip condition are always larger than for the full slip ones. In comparison with the incremental normal forces of the circular lining, incremental normal forces in the sub- rectangular lining are lower by about 9% and 25% for no-slip and full slip conditions, respectively (Figure 2.17b). a) Incremental bending moments b) Incremental normal forces Figure 2.17. Effect of the lining thickness on the incremental internal forces of the circular and sub-rectangular tunnel linings 2.5. Conclusion A 2D numerical study allowed investigating the behavior of sub-rectangular tunnel linings under seismic loadings. The influences of parameters, like the soil deformation, the maximum horizontal acceleration, the lining thickness, and soil- lining interface conditions, on the circular and sub-rectangular shaped tunnel behavior under seismic loading, were investigated. Considerable differences in the -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ex tr em e In cr em en ta l B en di ng M om en t M (M N m /m ) Lining thickness (m) Mmax_SR_ns Mmax_SR_fs Mmax_Circular_ns Mmax_Circular_fs Mmin_SR_ns Mmin_SR_fs Mmin_Circular_ns Mmin_Circular_fs -2 -1.5 -1 -0.5 0 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ex tr em e In cr em en ta l N or m al F or ce s N (M N /m ) Lining thickness (m) Nmax_SR_ns Nmax_SR_fs Nmax_Circular_ns Mmax_Circular_fs Nmin_SR_ns Nmin_SR_fs Nmin_Circular_ns Mmin_Circular_fs 49 response of these tunnels were observed. Based on the research results, conclusions can be deducted as follows: - The horizontal acceleration aH, soil’s Young modulus Es, and lining thickness t have a great effect on the incremental internal forces induced in both sub-rectangular and circular tunnels for both no-slip and full slip conditions; - In general, a higher seismic loading induced by a higher horizontal acceleration aH, will induce higher incremental bending moments and normal forces in both circular and sub-rectangular tunnels. The relationship is quite linear; - The results proved that the soil-lining interface conditions have a great influence on the behavior of sub-rectangular tunnels. This is completely different when comparing the behavior circular-shaped tunnels. Indeed, while the absolute extreme incremental bending moments of a circular tunnel for the no-slip condition are smaller than the corresponding full slip ones, the absolute extreme incremental bending moments of sub-rectangular tunnels for the no-slip condition are greater than the corresponding full slip ones. That is opposite to the trend observed in circular tunnel linings; - For all investigated case studies, absolute incremental normal forces for the no-slip conditions are always larger than the full slip ones, for both circular and sub- rectangular tunnels cases. Absolute extreme incremental normal forces in sub- rectangular tunnels are approximately 9% smaller than the circular ones; - The dependency of the absolute extreme incremental bending moments induced on sub-rectangular tunnels on the soil’s Young modulus (Es) is insignificant compared with the circular ones. Soil’s Young modulus of 50 MPa could be considered as a critical value for both tunnel shape cases. Beyond this value, the (Es) increase induces a decrease of the absolute extreme incremental bending moments. However, below this value, an increase in (Es) value induces an increase of the absolute extreme incremental bending moments; - An increase of the soil’s Young modulus (Es) causes a significant corresponding increase of the absolute extreme incremental normal forces for both 50 sub-rectangular and circular tunnels (no-slip condition). An insignificant change of the absolute extreme incremental normal forces is observed for the full slip conditions. The numerical results obtained in the present study are useful for the preliminary design of circular and sub-rectangular shaped tunnel linings under seismic loadings. These results also will be used in the next content in chapter 3. The joint distribution influence, in the segmental lining on the tunnel behavior, will be considered in a future research. 51 CHAPTER 3 A NEW QUASI-STATIC LOADING SCHEME FOR THE HYPERSTATIC REACTION METHOD - CASE OF SUB-RECTANGULAR TUNNELS UNDER SEISMIC CONDITION The HRM method was used to compute the structural forces developed in the tunnel lining [47],[48],[51],[52],[53],[92],[121],[122]. The HRM method developed using Matlab software provides a fast and accurate prediction of the internal forces in tunnel lining. The HRM method allows performing the calculations in a very short time manner, thus it is recognised to be appropriate for optimization design processes [92],[145]. The HRM method was successfully applied to estimate the seismic-induced structural forces in a circular tunnel lining considering pseudo-static condition [47],[145]. Based on the in-plane shear stresses applied on the tunnel lining proposed by Peinzen and Wu [128] and Naggar et al. [112],[114], Do et al. [47] applied a set of dimensionless parameters to change the external loading magnitude of the seismic- induced shear stresses. After Do et al. [47], Sun et al. [145] additionally considered the ground-tunnel interaction effect on the applied external loadings by using a dimensionless factor to