Structural design verifications, structural analysis of tunnels - Tunnel Engineering: Basics and Additional Services for Design and Construction - page 120

are determined through the use of the elasticity theory, however, these properties are not
considered in the calculation. Only when crack formation is considered with the resulting
loss of stiffness in the parts of the tunnel lining subjected to bending can the effect of re-
distribution of forces into less highly loaded parts of the structure and increased activation
of the subgrade reaction be included in the calculation.

Fig. 3-19 shows an example of the qualitative effect of an assumption of reduced bending
stiffness on the resulting moments in a two-track rail tunnel cross-section. The rotations
resulting from the introduction of plastic hinges are damped by the increased activation
of subgrade reaction by the ground around the tunnel. The result is a redistribution of the
bending moments with a simultaneous increase of the subgrade reaction.
The weakening of the system in areas of high bending action is associated with the as-
sumption of deformation, which has to be limited to a permissible degree. Steel fibre
concrete is indeed a markedly ductile construction material in comparison to unreinforced
concrete, but its plastic deformability cannot be assumed as limitless. Eurocode 2 [67]
requires that processes that deviate from the elasticity theory to determine the forces and
moments in the section of reinforced concrete structure also include a verification of the
plastic rotations required to redistribute the bending moments. No standardised proce-
dures for the verification of the plastic rotation capability of steel fibre concrete are yet
available (Fig. 3-19).
Simplified verification procedure for shotcrete tunnel outer linings. In tunnelling
practice, there is a verification procedure for tunnel outer linings of shotcrete, which ap-
proximately considers the mechanism of moment redistribution described above, and is
based on the investigations of Schikora/Ostermeier [200]. The basis of this procedure is
the calculation formula from DIN 1045 [53] for unreinforced concrete under eccentric
compression loading (1).
N ≤ b R · d / 2,1 · (1 - 2 · e / d )
N normal compression force
d thickness of the cross-section
b R calculation value of compression strength according to DIN 1045
e eccentricity of the normal compression force
According to the modified verification procedure of Schikora/Ostermeier [200], bending
moments are not necessary for the equilibrium of forces due to the static indeterminacy
of a bedded tunnel lining. They can be interpreted as constraint loading and then do not
require the application of the full factor of safety. Returning to an investigation for an
underground railway cross-section in loose ground based on the FE method, taking into
account the possible plastification of a shotcrete tunnel lining, the following calculation
is recommended:
N ≤ b R · d / 2,1 · [1 - 2 · ( e a + e ) / ( d · 2,1)]
e a additional eccentricity to take into account imperfections

The application of this simplified calculation formula for the verification of a tunnel outer
lining assumes, like the application of the more precise non-linear procedure, the plastic
rotation capability of the shotcrete or steel fibre shotcrete. Strictly speaking, this requires
an appropriate verification.