Unreinforced Concrete Lining For Railway Tunnel –
Nonlinear Analysis Based Design
Keywords: Tunnel lining, plain concrete
structure, nonlinear analysis, design methods, numerical simulation, finite
element method, concrete material models, ATENA
software.
Final lining
of the double-track railway tunnels (Fig. 1) was designed in both
reinforced and plain concrete. Since the application of unreinforced concrete
for tunnel lining is rather novel in
Design of
the plain concrete tunnel lining was supported by nonlinear finite element
analysis. For the analysis the commercial software package ATENA was used. This is a well established finite element program for realistic computer
simulation of damage and failure of concrete and reinforced concrete structures.
In ATENA special constitutive models
for concrete are employed:
· Tensile behaviour of concrete is
modelled by non-linear fracture mechanics combined with crack band method and
smeared crack concept. A real discrete crack is simulated by a band of
localized strains. Tensile softening (i.e. drop of tensile stresses after
crack initiation) is assumed in an exponential form and is driven by fracture
energy consumption.
· Concrete in compression is covered
by special theory of plasticity with a non-associated flow rule and compressive
softening damage. It can reflect influence of lateral stresses to the
compressive strength, so called confinement, which can play an important role
in 3D and plane strain situations.
· Steel reinforcement can be
introduced into concrete in different ways: discrete bars, pre-stressing
cables, or as a smeared reinforcement. The discrete reinforcement can have any
arbitrary geometrical form and is fully independent on the finite element mesh.
It can be assumed with perfect bond to the surrounding concrete, or a bond-slip
law can be employed. For the smeared reinforcement the reinforcing ratio in
certain part of the structure can be prescribed.
Fig. 1Tunnel cross section
The
efficient numerical solution of the structural model in ATENA is accomplished by user-friendly graphical environment. It
supports preparation of input data for the nonlinear finite element analysis,
enables real-time graphical tracing and control during the analysis, and allows
an efficient evaluation of results. In particular, the graphical post-processing
can show cracks in concrete with their thickness and residual stresses;
realistic crack patterns can be obtained. All the important values from the
analysis (strains, stresses, deflections, internal forces, reactions etc.) can
be represented graphically as coloured iso-areas, diagrams in defined
cross-sections, or in form of vector or tensor arrow plots. In addition to the usual
finite element results, engineering integral forces (i.e. bending moments,
normal and shear forces) can be calculated and plotted.
A tunnel
section of 1 m length was modelled in ATENA
2D in plane strain state. External
diameter of the upper tunnel tube was 6 m. The structure was supported by
springs around the tube and under the abutment; spring stiffness was derived
from ground properties. Various lining alternatives were considered and
analyzed: concrete shell thickness of 0.300 m or 0.350 m, plain or reinforced
concrete, without or with a bottom vault. An alternative with fixed abutment
has been analyzed as well. The tunnel structure was loaded by dead load, creep
and shrinkage, temperature in summer or winter conditions, and ground pressure.
All the desired load combinations were calculated and evaluated individually,
since superposition of the load cases is not possible in the nonlinear
analysis.
Material
properties for the concrete tunnel shell were considered in three alternatives,
since requirements of the codes had to be proved:
The nonlinear
finite element analysis performed with the realistic material values simulated expected
real structural behaviour. From the calculated overloading compared to the
design values global safety of the structure was assessed. The calculation with
characteristic values was used for serviceability limit state evaluation (crack
width, deformations, abutment safety). In case of the analysis with design
values the loads were enlarged by load partial safety factors. Results from
this calculation were used for verification of the ultimate limit state in M-N
interaction diagram.
As example
of graphical outputs from the ATENA nonlinear
analysis are presented several results for the unreinforced lining of
0.300 m thickness, without bottom vault, with spring supported abutment, in
the load combination B – dead load + creep and shrinkage + temperature in
winter. Crack pattern for the ultimate limit state with a detail of the main
crack in the upper vault are shown in Figures 2 and 3. Bending moments and
normal forces along the cracked tunnel lining are drawn in Figure 4.
Fig. 2Crack pattern in the ultimate limit state
Fig. 3Detail of the main cracks in the upper vault with crack widths description
Fig. 3Bending moments and normal forces along the cracked lining
Unreinforced
tunnel lining was designed with support of nonlinear finite element analysis
using ATENA software. Both these
methodologies, i.e. tunnel lining made from plain concrete, and nonlinear
analysis based design, are rather novel, at least in