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(2.16)
Note that when , , , Coulomb’s earth pressure coefficient becomes equal to , which is the same as Rankine’s earth pressure coefficient given earlier in this chapter.
The variation of the values of and with and can be given by equations (2.14) and (2.16), respectively. The value of and increases with the wall friction. Note that making the assumption that the failure surface is a plane in Coulomb’s theory grossly reduces somewhat the active earth pressure coefficient and overestimates the passive resistance of walls. Especially for , this error that overestimates the passive resistance is somewhat unsafe for all design purposes.
2.3 Rock Mass Pressure
The earliest reference to the use of rock loads based on the rock mass classification for the design of tunnel support is in a paper by Terzaghi(1946). And Terzaghi’s descriptions are as follows:
▲ Intact rock contains neither joints nor hair cracks. Hence, if it breaks, it breaks across sound rock and needs light lining.
▲ Stratified rock consists of inpidual strata with little or no resistance against separation along the boundaries between the strata. In such rock the spalling condition is quite common and light support required and load may change erratically from point to point.
▲ Massive and moderately jointed rock contains joints and hair cracks, but the blocks between joints are locally grown together or so intimately interlocked that vertical walls do not require lateral support. Load may also change erratically from point to point.
▲ Moderately blocky and seamy rock consists of chemically intact or almost intact rock fragments which are entirely separated from each other and imperfectly interlocked. And little or no side pressure.
▲ Completely crushed but chemically intact rock has the character of crusher run and require to considering side pressure. Softening effect of seepage toward bottom of tunnel requires either continuous support for lower ends of ribs or circular ribs.
▲ Squeezing rock slowly advances into the tunnel without perceptible volume increase. And invert struts required with heavy side pressure. Most of conditions for such rock, the circular ribs are recommended.
▲ Swelling rock advances into the tunnel chiefly on account of expansion. In such rock circular ribs are required and in extreme cases the yielding support is needed.
If the support structures used to ensure the stability of the opening is relatively stiffer than the host ground, the support structure will attract more loading. In the same situation, a support system that is more flexible than the host ground will take lesser load than a stiffer support. In case of a flexible support, the ground by arching will take the major portion of the load and the support system will take a smaller share of load. A stiffer support attracts more load and a flexible support attracts more displacement. A steel support is more flexible than a concrete lining.
Figure 2.4 indicates a ground characteristic curve in which the ground pressure is plotted as an ordinate and radial displacement as an abscissa.
Figure.2.4. Characteristic Curve
At time “ ”, the theoretical pressure on support is “ ”, the in-situ pressure and the radial deformation “ ” is zero. Theoretical, because it is impossible to place a support without relaxing the ground and without reducing “ ”. When an opening is created, the excavation moves toward the opening and the value of “ ” starts to diminish. The portion “AB” of ground characteristic curve is purely elastic. From “B” to “C”, the ground starts to yield, but by “arching”, it can still take some load. From “C” onward, the ground starts to “loosen” and it can no longer sustain any load. At time “ ” when a support is placed to arrest the radial movement, it will have to sustain a pressure equal to “ ”. If the same support is placed at time “ ”, it will have to sustain a load equal to“ ”. As can be seen, “ ” is smaller than “ ”. A prudent design will be to place the support at time “ ” or just before the ground starts to loosen itself. At that time, the support will be required to sustain the least pressure “ ” to keep the opening stable. It is, however, very difficult to assess the exact time “ ” after which the ground starts to loosen up (R.S.Sinha, 1989).