Structural design verifications, structural analysis of tunnels - Tunnel Engineering: Basics and Additional Services for Design and Construction - page 105

3.4.4 Bedded beam models
The interaction between ground and tunnel construction is simulated in this model by the
so-called bedding or stiffness modulus procedure. The tunnel is represented by a non-
linear bedded beam and is wholly or partially surrounded by bedding. A subgrade reaction
is activated as soon as deflection into the ground occurs, which simulates the load-bearing
effect of the ground. A certain proportion of the load is resisted by the subgrade reaction,
controlled by the modulus of subgrade reaction. The tunnel construction can be repre-
sented in analytic calculation methods as an elastically bedded non-linear beam in the
elastic continuum or if the geometry is more complex, as a bedded continuous beam ring
in a numerical calculation procedure. Bedded beam models can also be used to model the
geometry of cross-sections with partial areas that are not embedded. The numerical model
of a bedded beam is frequently used in continuous beam programs specially modified for
tunnel design.

The modulus of subgrade reaction is not a soil constant but depends on the modulus of
elasticity of the ground E g , the radius of the tunnel R and the factor C . Usual values for C
lie between 0.5 und 3.0 [4, 160]:

CE
R

⋅

g

k =

s

In addition to the full-round continuously bedded system, partially bedded systems are
also used. In this case, a 90° unbedded crown segment is typical, since the support in most
cases deforms inward in this zone. For the calculation, care should generally be taken that
displacements into the tunnel do not activate any tension forces in the bedding, since in
reality the ground will only provide a subgrade reaction in compression. The value for the
bedding around the perimeter does not have to be constant but can change linearly or in
steps, particularly for systems with unbedded crown. Radial or tangential bedding can be
differentiated according to the direction of the subgrade reaction. While the assumption of
radial bedding is always realistic, the applicability of tangential bedding in contrast must
always be checked. This applies particularly to calculations modelling the inner lining of
two-layer constructions separated by a foil.
Depending on the depth of the tunnel, the vertical overburden pressure can be assumed
to act fully, or for deep tunnels partially reduced, as an external load. The horizontal
load component is determined from the appropriate coefficient of lateral ground pres-
sure. When a bedded beam model is used, different variants are available depending on
the beam program used. The optimal representation of the tunnel geometry is as an arch
consisting of curved members. It is also however possible to approximate as a polygonal
continuous beam. In case no continuously bedded beams are available, the bedding can
also be represented by individual radial and tangential springs or hinge-ended columns at
the node points (Fig. 3-8).
Even at a time when complex calculation programmes are available, the non-linear bedded
beam is still a useful model for the design of shallow tunnels in loose ground. Because of
its relatively simple manageability, the processing time is short compared to other numeri-
cal processes. One disadvantage is the limited extent of the results, which are restricted to
the internal moments, forces and deflections in the members and the subgrade reactions.