Seismic Ovalization Analysis of Tunnel Permanent Linings

Question

Hello to all, 
 We are performing seismic ovalization analysis to calculate stresses on tunnel permanent linings which will be excavated in Andesite Rock Formations, Parameters of the rock formation are as follows: 
 GSI= 40 
 UCS= 30 MPa 
 mi= 25 
 Erm= 2400 MPa 
 First we have done the calcualtions with analytical methods proposed by Wang and assumed no slip condition between tunnel lining and rock for axial loads and full-slip condition for momment and shear force. 
 Then we have done ovalization analysis on plaxis to compare and verify the results with analytical formulations , we modeled the rock formation as lineer elastic material and assuming again no slip condition between tunnel lining and rock for axial loads and full-slip condition for momment and shear force. 
 We have concluded similar results between analytical method and plaxis ovalization analysis. But there was a problem, we saw from the results that the axial tension loads came from seismic calculations are way too much from static compression axial loads from static analysis, and when we add this forces to seismic axial loads, it leads to axial tension forces around 2000 kN, which we were not expect such amount of tension. 
 The interaction diagram of the tunnel liningis as follows:

Then we tried to re-model the plaxis model with a non-lineer material model such as Mohr-Coulomb model to see the difference and we saw that there were no axial tension load on tunnel lining for MC model. and the amount of axial load on compression was similar to linear elastic model 
 
 Below you can see the results of both cases: 
 
 These results make confusions on reinforcement design of the lining, the regulations which we have to follow are suggesting to use analytical methods, but it is resulting too much axial tension on lining. 
 We are thinking that especially for this kind of good rock can act linear-elastic while seismic event. but the axial tensions force on lining taking us to put too much reinforcement in lining. 
 Are there anyone who experienced the same results? and how to solve it? 
 Thanks in advance for the helps

Berk Demir · Answer

Hi Osman, 
 I have been on this position before. I know what you are experiencing. The problem is caused by following: 
 You have two loads - static and seismic. When you are calculating the seismic load for your reinforcement calculation, you actually take both static and seismic into account. We know that seismic load creates tension due to ovalization. You can double check that by Wang or comparing the axial forces after the seismic forces in Plaxis. Seismcity creates tension. 
 The problem is, when you are on a soft soil, even though the ovalizations are high (and tension is also high), the resulting axial force does not go negative, because the axial force due to static forces are high. So, in the end you end up with compression, which is very favourable. 
 But in very hard rock, the static axial force is very low because rock is very good. Therefore, when you apply high seismicity, the ovalization creates tensile forces that exceed compression forces. 
 But, there is another problem with your model. Modelling rock as linear elastic material gives the material infinite tensile strength, which creates more ovalization forces. This is in-line with no-slip assumption, but in reality rock has a limited (though very high) tensile strength. However, as far as I see this is a NATM tunnel, so maybe you can consider lower Rinter values to relief the axial forces if you use Hoek Brown model. 
 With MC, you get no forces, because when it try to ovalize, the rock-tunnel interface just slips, because you probably didn't enter any tensile capacity to the model. So, it's completely different from what you have in linear elastic model. 
 This is a fun and niche part of tunnel engineering, enjoy!

Berk Demir · Answer

I assume with Hoek Brown model, even with Rinter = 1.0, you should not have high tensile forces, because tensile strength of rock is very small as you stated. So, for both no-slip and full-slip case, you are calculating very low axial forces. 
 Don't confuse no-slip / full-slip with low tension capacity of the rock. If you use Rinter=1.0, that means no-slip in FEM. So, that will also ease your job. 
 I suggest you go with Hoek-Brown and Rinter=1.0 & Rinter=0.1, check to see which governs, and note that in your report. I think, if you are talking about standards of Road Directorate of Turkey, this should be under 6.4 - numerical methods. Because, if you do otherwise, you will end-up thinking and thinking. But if you crunch the numbers, you will see that there is around 60 times difference between axial forces. Compression is OK, but since you have to use this force also as tension due to sin(2*theta) thing, that means also tension. 
 
 The hand calculations are extremly overconservative - even though I used them to do some checks of my calculations before. It's not the Turkish Directorate that puts that requirement actually. It's Wang's recommendations. Whole standard is the translation of Wang's and Hashash's works. (See the second screenwhot below.) 
 
 Anyhow, we definitely exceeded the purpose of the forum. So, I suggest you keep going with the methodology I recommended and never use linear elastic soil material in these types of analyses. Only use when the strength does not matter - like purely settlement analysis.