1. Structural Analyses of Segmental Lining – Coupled Beam and
Spring Analyses Versus 3D-FEM Calculations with Shell Elements
C. Klappers, F. Grübl, B. Ostermeier
PSP Consulting Engineers for Tunnelling and Foundation Engineering, Munich, Germany
ABSTRACT
In contrast to the inner lining of a NATM tunnel the lining of a TBM driven tunnel consists of single
precast concrete segments which are articulated or coupled at the longitudinal and circumferential
joints. Therefore not only the characteristics of the concrete segments influence the structure but also
the mechanical and geometrical characteristics of the joints strongly affect the structural behaviour of
the tunnel lining. For the simulation of these joints within the tunnel lining different calculation
methods are known.
In the following it is shown how the behaviour of the joints can be modelled in an appropriate
way. Different calculation methods with beam and spring models and 3D-FEM models are compared
and discussed. It can be seen, that for the structural design of the segments for regular cases
calculations with special beam and spring models are sufficient whereas 3D-FEM calculations are
necessary when the spatial bearing behaviour of the lining with respect to the bearing behaviour of the
joints needs to be considered.
1. INTRODUCTION
Currently beam and spring models (BSM) analysis with coupled, hinged rings can be considered as
state of the art model for the structural design of a segmental lining. However, in special cases such as
openings in the lining for cross passages, BSM analysis do not provide reliable results since the
structural behaviour of the tunnel lining in longitudinal direction, the deformation of the lining due to
the rotation in the longitudinal joints and the relative displacement in the circumferential joints have to
be taken into account. These effects can be
simulated with 3D-FEM calculations with
bedded shell elements connected with non-
linear springs, representing the rotational
stiffness of the concrete hinges in the
longitudinal joints and the coupling of the
segmental rings in the circumferential joints.
The different calculation approaches for
the structural design of a segmental lining are
described and for a typical configuration of a
segmental lining the results of BSM analysis
with coupled, hinged rings are compared with
the results of 3D-FEM calculations.
Figure 1. Segmental lining
1

2. 2. STRUCTURAL DESIGN FOR A SEGMENTAL LINING WITH BEAM AND SPRING
ANALYSES
All calculations mentioned in this paper base on a reference
tunnel with a system radius of 5.1 m, 40 cm segment thickness,
2 m ring length, oedometric modulus of 150 MPA, vertical load
of 250 kPa and Ko=0.6. Each ring is built of 6 segments.
Modells are given by two rings in general (ring 1 and ring 2).
Two systems are examined. At system I ring 1 has no
hinge at the crown and ring 2 is rotated by half a segment
which means that there is a hinge at the crown. This is the most
unfavourable configuration in terms of the bending moment at
the crown. At system II all hinges are rotated by 15° compared
to system I.
2.1 Different structural systems Figure 2. Ring configurations
First of all it has to be differentiated between coupled or uncoupled segmental rings. A lining built
with straight longitudinal joints behaves as uncoupled hinged ring, whereas in systems built with
staggered joints the rings interact and the distribution of the internal forces is changing.
There are a lot of different structural systems known in the design practice to calculate the
internal force within the tunnel lining. The most simple one is to use a rigid bedded ring. This model
does not take the behaviour of the joints into account. For an uncoupled system of hinged rings the
estimated bending moments are too high and should give conservative results. Sir Allan Muir-Wood
(1975) developed a very easy to use empirical formula to estimate the effects of the longitudinal joints
of uncoupled rings in a calculation with a homogenous rigid ring by reducing the bending stiffness of
the lining. The maximum bending moments calculated with this approach are quite close to the
maximum bending moment calculated for a hinged uncoupled ring. For coupled rings these moments
are mostly to small, especially with a configuration like system I. However, this approach is quite
useful to get a first idea of the forces in the lining.
To calculate the internal forces of a segmental lining with staggered joints in a proper way it is
essential to simulate the coupling in the circumferential joints. Therefore bedded BSM analyses with
coupled, hinged rings are very common for the structural design of a segmental lining.
In all of the following calculations the beams are bedded with non linear radial springs which do
not allow tension forces. The assumptions for the behaviour of the joints are done for plane
longitudinal and circumferential joints, because in many cases the use of tongue and groove or other
types of mechanical coupling is deemed to be not necessary or useful .
2.2 Bedded beam and spring model analysis with coupled, hinged rings
As the characteristics of the joints are essential for the structural behaviour of the system the
mechanical properties of these joints have to be simulated
in an appropriate way. 200
Longitudinal joints: For the determination of the 150
rotational stiffness of the longitudinal joints usually the
C m [MNm/rad]
100
formulas from Janssen (1983) based on the investigations 50
of Leonhardt and Reimann (1966) for the resistance 0
against rotation and bending of concrete hinges are used. -0,15 -0,1 -0,05
-50
0 0,05 0,1 0,15
As long as the joint is completely compressed the M [MNm]
rotational stiffness cm is constant and could be described
E ⋅ b² Figure 3. Relation of bending moment
as cm= . It depends only on the young’s and rotation stiffnes
12
modulus E and the width b of the contact zone. If this
2

3. bending moment exceeds the boundary bending moment Mbou < N . b / 6 the joint is opening like a
bird’s mouth. From this point the rotational stiffness depends on the normal forces N and the bending
moment M and is described as
9⋅ E
c M= ⋅ ( 2 ⋅ M − N ⋅ b)³ (1)
32 ⋅ N ³ ⋅ b
For the implementation of this behaviour the non linear rotational springs should be able to fulfil
the above mentioned relationship between bending moment and rotation stiffness. It is not necessary
to define a yielding moment because the spring becomes extremely soft if the moment increases to
more than about 80 % of the maximum moment. If only a linear rotational spring with the definition of
a yielding moment is used the estimation of behaviour of the joint seems to be too rough.
Circumferantial joints: The coupling of the rings is simulated by lateral springs. In literature there
is not very much published about the modeling of the coupling between the rings. Usually the
coupling of the rings is simulated by using non linear lateral springs which represent the shear
stiffness and the maximum bearing capacity of the coupling. When using a plane joint with plywood
hardboards the spring stiffness is given by the shear stiffness of the plywood c=, where G is
representing shear modulus, A is the area of hardboard and d is the thickness of hardboard.
Even without a mechanical coupling the rings are coupled by friction between plywood and
concrete. This is caused by forces in the circumferential joint due to the influence of the hydraulic
shoving rams of the TBM. The value of the frictional coefficient µ is hard to define and is subject of
discussions. At laboratory tests which were undertaken for the 4th Elbtunnel Hamburg from STUVA
(1996) µ =0.25-0.3 was discovered. Gijsbers and Hordijk (1997) did similar tests for tunnel projects in
the Netherlands. For plywood hardboards they found µ =0.4-0,7 as friction coefficient. After reaching
the maximum force the residual friction coefficient decreased to µ =0.3-0.55. The minimal values for
µ were found for normal stresses of about 35 MPA at the hardboards and maximum values for normal
stresses of about 12 MPA. Because of the limited compressive WI NGRAF ( V13. 61 -2 1) 1 5 .10 .2 005 PSP Bera tend e Inge nieu re
00
6.
strength of concrete normally the area of the hardboard will be
chosen big enough that the normal stress at the hardboards will be
00
4.
less than 20 MPA. Approximately they will be around 10 an 20
.00
MPA. All these tests were done in laboratories with unbedded
2
concrete segments where the segments could move independently
0.00
from each other. Due to the grouting of the tail gap and the
surrounding ground the deformation of the segments is harmonized
in real conditions on site. -2
-4.00
.00
In the structural analysis the radial springs which simulate the
bedding of the rings can also deform independently. Therefore the
-6.00
4 0
.0 2. 00 0.00 - 2.00 -4.00 m
effect of harmonized deformation has to be considered when
Y Str uktur M 1 : 35
X X * 0.819
Z Y * 0 9
.9 6
Z * 0.581
Spring- Be am cou pled R ng Rsy s
i =5.1 file :f ull _coup_ V40
choosing the frictional coefficient for the coupling springs. Figure 4. Structural system of
Because of the above mentioned matters taking µ=0,5 into account the coupled spring beam model
seems to be a reasonable value. It will be used in the following
calculations. For structural final design the value of µ should be varied. Within the analysis the
maximum bearing capacity of the lateral springs depends on the chosen frictional coefficient and the
applied shoving forces. The whole system of the model for the BSM analysis consists of two half rings
(with respect to the ring length) coupled with the above mentioned lateral springs.
3

4. Table 1. Results for different structural systems
system I system II
Structural system rigid ring Muir- uncoupled uncoupled coupled coupled uncoupled uncoupled coupled coupled
Wood ring ring 1 ring 2 ring 1 ring 2 ring 1 ring 2 ring 1 ring 2
max bending 157 132 150 95 206 115 131 122 178 152
moment [kNm/m]
percentage 119% 100% 114% 72% 156% 87% 99% 92% 135% 115%
max settlement at 9 9,9 9 11,6 9,5 9,6 9,5 9,6 9,3 9,3
crown [mm]
percentage 91% 100% 91% 117% 96% 97% 96% 97% 94% 94%
As table 1 shows the calculation with a rigid ring does not give the maximum bending moment. The
bending moments for the coupled rings are always higher. The calculation with the reduced stiffness
according to Muir-Wood fits very well to the uncoupled calculations of system II. The coupled
calculations show that ring 1 of systems I behaves much stiffer than ring 2 which causes a load
transfer from ring 2 to ring 1. This leads to a much higher bending moment at the crown of ring 1.
These results demonstrate that for the given loads the ring configuration of system II is more
favourable for the design of the lining. The coupling of the rings reduces the deformation, but
increases the bending moments especially for the “stiffer” ring. From this calculation it can be seen
that for the final design at least for the critical load cases, BSM analyses with coupled, hinged rings
shall be done. With models which are more simple the bending moments might be underestimated.
3. CALCULATION WITH A 3D-FINITE-ELEMENT-METHOD (FEM) MODEL
In comparison to calculations mentioned in chapter 2 also calculations with a 3D-FEM-program
(prepared by SOFiSTiK) were done to check the quality of the results from the BSM analyses.
3.1 Modelling of the structure
For the 3D-FEM calculations the tunnel was modelled by a sufficient number of complete rings. The
ring configuration is taken as described above in system I. The segments are modelled with plane 4-
node shell-elements with a non-conforming formulation. These elements can be bedded in radial and
tangential direction. For the bedding non linear effects like failure, yielding and friction can be
defined. Each segment consists of 5 elements in longitudinal direction an 18 elements in tangential
direction which means 540 elements per ring. At the longitudinal joints the adjacent segments are
coupled with 6 rotational springs. In the circumferential joints the segments are coupled with 3 lateral
springs per hardboard which means 72
springs per joint. The mechanical, non-linear
properties of the different springs are the
same as for the BSM analysis described in
chapter 2.2.
Since the maximum possible coupling
forces depend on the shoving forces of the
TBM the calculations were done for a variety
of total shoving force between 40 to 5 MN.
Figure 5. 3D-FEM-structure 3.2 Comparison of the results of the spring
beam model and the
3D-FEM Model
With the 3D model
coupled and uncoupled
systems were
calculated. In the
figure 6 the effects of
4
uncoupled coupled
Figure 6. Deformed structures (scaled up)

5. the coupling of the rings are obvious. At the uncoupled system each ring deforms independently and at
the coupled system the deformation of the rings is harmonized.
BSM with coupled rings 3D-FEM
Structural system uncoupled uncoupled coupled coupled uncoupled uncoupled coupled coupled
ring 1 ring 2 ring 1 ring 2 ring 1 ring 2 ring 1 ring 2
crown bending 150 95 206 82 155 95 201 82
moment [kNm/m]
max settlement at 9 11,6 9,5 9,6 9,1 11,2 9,1 9,3
crown [mm]
Table 2. Comparison of the results of the beam and spring and the 3D-FEM model
A comparison of BSM and 3D-FEM model shows, 250
that the calculated bending moments of both models are in 230
crown bending moment [kNm]
a similar range and deformations differ only slightly. The 210
190
deviation of the bending moments calculated with various 170
BSM ring 1
total shoving forces is only about 5%. This is because the 150 BSM ring 2
coupling forces which are necessary to harmonize the
3D-FEM ring 1
130
3D-FEM ring 2
deformation of the rings are very small. If a total shoving 110
90
force of more than about 5 MN is applied to the system it 70
behaves like the rings were fully coupled. The applied 50
0 5 10 15 20 25 30 35 40
shoving force will become more effective to the system if advance force [MN]
for example the loads are not equally distributed. 12
For usual cases where the loads and the structure BSM ring 1
does not change in longitudinal direction the three-
crown settlement [mm]
11 BSM ring 2
3D-FEM ring 1
dimensional structural behaviour of the segments has no 3D-FEM ring 2
significant influence to the system. That means for this 10
kind of load configurations 3D-FEM calculations are not
9
necessary. For special cases like openings in the lining,
different loads on the rings (e.g. swelling only in partial 8
areas), varying bedding conditions for the rings (e.g. if the 0 5 10 15 20 25 30 35 40
advance force [MN]
grouting of the tail gap was not done properly at one ring)
or other special cases only with 3D-FEM calculations the Figure 7. Bending moment and
internal forces and deformations of the lining can be crown settlement
predicted in a serious way.
3.3 Segmental lining with an opening and a temporary bracing
Very often the segmental lining has to be opened to build cross
passages between two tubes. During the advance of the passage
tunnel it is usual to install a steel framework at the running
tunnel before opening the window. The bearing behaviour of
such a structure with a slender steel frame around the opening
was analysed with the 3D-FEM model.
Figure 8. Deformed structure
5

6. The steel framework is build of rigid beam elements. The horizontal beams are connected to the
segments with hinges. The stems are connected to the segments with springs which can only transfer
compression forces. With 25 250
respect to the excavation of 230
crown bending moment [kNm]
210
the cross passage, the bedding
20
190
deformation [mm]
stiffness around the window is
AUTHOR : PSP Beratende Ingenieure 80686 München
15 170
PROGRAM : WINGRAF VERSION 13.61-21 (c) SOFiSTiK AG BSM ring 1
reduced and the maximum
crown settlement 150 BSM ring 2
PROJECT : 3D coupled Ring Rsys=5.1 file:3D_opend_coup_V15 ASB NO. : DATE :
3D-FEM ring 1
10 23.10.2005
differential 130
3D-FEM ring 2
bedding stress is limited to the deformation
110
uniaxial compressive strength 5 90
70
of the surrounding ground. 0 50
5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
advance force [MN] advance force [MN]
-6.00
Figure 9. Bending moments and deformations of opened ring
These calculations show that if the total of shoving forces
become smaller than 20 MN combined with reduced possible
-4.00
coupling forces the maximum bending moments and the
deformations start to increase rapidly. Especially at the invert the
bending moment increases about 80 % and the differential radial
deformation at the circumferential joints becomes more then 5
-2.00
mm. It can also be seen that with the chosen kind of bracing a
minimum coupling between the rings is needed. If the shoving
forces become less than 5 MN the investigated system starts to 0.00
become unstable. With simulations like this it is possible to
calculate the bearing capacity of the opened lining. It is also
possible to define a minimum shoving force which has to be used during the shoving of the tunnel at
the area of the cross passage or maybe to decide that another kind of bracing is necessary.
2.00
Figure 10. Stress distribution around the opening
4.00
4. CONCLUSIONS
From the shown calculations it can be seen that the structural behaviour of the joints must be taken
6.00
into account within the structural analysis of the segmental lining. For normal load cases beam and
spring analyses with coupled hinged rings are sufficient. In special cases were the 3D bearing
behaviour of the whole tunnel has to be considered
2.00
FEM calculations with bedded shell elements give a
4.00 6.00 8.00 10.00 12.00 14.00 m
good impression of the internal forces and the
Sector of system M 1 : 75
deformations of the system. For all types of calculations
X Z
Plane Principal stresses in Nodes, nonlinear Loadcase 1 GEBIRGSDRUCK+QUELLDRUCK, 1 cm 3D = 7.81 MPa
Y
+= -= (Min=-18.0) (Max=5.97)
PART : the behaviour of the joints has to be modelled in a ARCHIV NO
BLOCK :
DETAIL : proper way, because these joints will highly affect the
results. The possible minimum and maximum coupling
forces have to be taken into account and a parametric
study with a variety of coupling forces shall be done.
Normally, the maximum coupling forces will give the
maximum bending moment and the minimum coupling
Figure 11. Deformed structure of segmental
forces will cause the maximum deformation. When the
lining with swelling loads at one ring
lining is opened to build a cross passage or a high
locally load has to be applied to a single ring, the bending moments will increase with the decreasing
of the possible coupling forces. In such cases a minimum amount of possible coupling forces could be
necessary to assure the stability of the whole system. Due to to the high efforts the shown 3D-FEM
calculations are not common practice. They should be reserved to cases needed.
6

7. REFERENCES
Sir Muir Wood, A.M., 1975, "The circular tunnel in elastic ground", Géotechnique 25(1)
Janssen, P., 1983, "Tragverhalten von Tunnelausbauten mit Gelenktübbings", Report-No. 83-41
University of Braunschweig, Department of civil engineering, Institute for structural analysis
Leonhard, F., Reimann, H.; 1966, ”Betongelenke”. Der Bauingenieur 41, p. 49-56
STUVA (editor), 1996, “Eignungsprüfungen 4. Elbröhre Elbtunnel, Reibungsversuche”,
www.stuvatec.de/tubbing_ergebnisse.htm
Gijsberg, F.B.J., Hordijk, D.A., 1997, “Experimenteel onderzoek naar het afschuifgedrag von
ringvoegen”, TNO-rapport COB K111
Grübl, F., 2005, “Ring Coupling for segmental Linings – old Hat or Necessity?, Tunnel special edition
IUT 05
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