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.

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Figure 1. Segmental lining

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

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.

Longitudinal joints: For the determination of the rotational stiffness of the longitudinal joints usually the formulas from Janssen (1983) based on the investigations of Leonhardt and Reimann (1966) for the resistance against rotation and bending of concrete hinges are used. As long as the joint is completely compressed the rotational stiffness cm is constant and could be described

as cm=12

²bE ⋅. It depends only on the young’s

modulus E and the width b of the contact zone. If this

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Figure 3. Relation of bending moment and rotation stiffnes

-50

0

50

100

150

200

-0,15 -0,1 -0,05 0 0,05 0,1 0,15

M [MNm]

Cm

[MNm

/rad]

Figure 2. Ring configurations

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

cM= )³2(³32

9 bNMbN

E ⋅−⋅⋅⋅⋅

⋅ (1)

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 strength of concrete normally the area of the hardboard will be chosen big enough that the normal stress at the hardboards will be less than 20 MPA. Approximately they will be around 10 an 20 MPA. All these tests were done in laboratories with unbedded concrete segments where the segments could move independently 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.

In the structural analysis the radial springs which simulate the bedding of the rings can also deform independently. Therefore the effect of harmonized deformation has to be considered when choosing the frictional coefficient for the coupling springs. Because of the above mentioned matters taking µ=0,5 into account 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.

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PSP Bera tend e Inge nieu reWI NGRAF (V13.61-21) 15.10.2005

Spring-Beam cou pled Ri ng Rsy s=5.1 file:full _coup_ V40

M 1 : 35X

YZ X * 0.819

Y * 0.9 96Z * 0.581

Struktur

4.0 0 2.00 0.00 -2.00 -4.00 m

-6.0

0-4

.00

-2.0

00.

00

2.0

04.0

06.0

0

Figure 4. Structural system of the coupled spring beam model

Table 1. Results for different structural systems

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.

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

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Figure 5. 3D-FEM-structure

uncoupled coupledFigure 6. Deformed structures (scaled up)

Structural system rigid ring Muir-Wood ring

uncoupledring 1

uncoupledring 2

coupled ring 1

coupled ring 2

uncoupledring 1

uncoupledring 2

coupled ring 1

coupled ring 2

max bending moment [kNm/m]

157 132 150 95 206 115 131 122 178 152

percentage 119% 100% 114% 72% 156% 87% 99% 92% 135% 115%max settlement at crown [mm]

9 9,9 9 11,6 9,5 9,6 9,5 9,6 9,3 9,3

percentage 91% 100% 91% 117% 96% 97% 96% 97% 94% 94%

system I system II

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.

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, that the calculated bending moments of both models are in a similar range and deformations differ only slightly. The deviation of the bending moments calculated with various total shoving forces is only about 5%. This is because the coupling forces which are necessary to harmonize the deformation of the rings are very small. If a total shoving force of more than about 5 MN is applied to the system it behaves like the rings were fully coupled. The applied shoving force will become more effective to the system if for example the loads are not equally distributed.

For usual cases where the loads and the structure does not change in longitudinal direction the three-dimensional structural behaviour of the segments has no significant influence to the system. That means for this kind of load configurations 3D-FEM calculations are not necessary. For special cases like openings in the lining, different loads on the rings (e.g. swelling only in partial areas), varying bedding conditions for the rings (e.g. if the grouting of the tail gap was not done properly at one ring) or other special cases only with 3D-FEM calculations the internal forces and deformations of the lining can be 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.

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Figure 8. Deformed structure

Figure 7. Bending moment and crown settlement

50

70

90

110

130

150

170

190

210

230

250

0 5 10 15 20 25 30 35 40

advance force [MN]

crow

n be

ndin

g m

omen

t [kN

m]

BSM ring 1BSM ring 23D-FEM ring 13D-FEM ring 2

8

9

10

11

12

0 5 10 15 20 25 30 35 40

advance force [MN]

crow

n se

ttlem

ent [

mm

] BSM ring 1BSM ring 23D-FEM ring 13D-FEM ring 2

Structural system uncoupledring 1

uncoupledring 2

coupled ring 1

coupled ring 2

uncoupledring 1

uncoupledring 2

coupled ring 1

coupled ring 2

crown bending moment [kNm/m]

150 95 206 82 155 95 201 82

max settlement at crown [mm]

9 11,6 9,5 9,6 9,1 11,2 9,1 9,3

BSM with coupled rings 3D-FEM

Figure 10. Stress distribution around the opening

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 respect to the excavation of the cross passage, the bedding stiffness around the window is reduced and the maximum bedding stress is limited to the uniaxial compressive strength of the surrounding ground.

These calculations show that if the total of shoving forces become smaller than 20 MN combined with reduced possible 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 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 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.

4. CONCLUSIONS

From the shown calculations it can be seen that the structural behaviour of the joints must be taken 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 FEM calculations with bedded shell elements give a good impression of the internal forces and the deformations of the system. For all types of calculations the behaviour of the joints has to be modelled in a 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 forces will cause the maximum deformation. When the 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.

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Figure 11. Deformed structure of segmental lining with swelling loads at one ring

AUTHOR :PROGRAM :

PROJECT : ASB NO. : DATE :

PSP Beratende Ingenieure 80686 MünchenWINGRAF VERSION 13.61-21 (c) SOFiSTiK AG

23.10.20053D coupled Ring Rsys=5.1 file:3D_opend_coup_V15

PART : ARCHIV NOBLOCK :DETAIL :

M 1 : 75XY

ZSector of systemPlane Principal stresses in Nodes, nonlinear Loadcase 1 GEBIRGSDRUCK+QUELLDRUCK, 1 cm 3D = 7.81 MPa+= -= (Min=-18.0) (Max=5.97)

2.00 4.00 6.00 8.00 10.00 12.00 14.00 m

6.00

4.00

2.00

0.00

-2.0

0-4

.00

-6.0

0

Figure 9. Bending moments and deformations of opened ring

0

5

10

15

20

25

5 10 15 20 25 30 35 40

advance force [MN]de

form

atio

n [m

m]

crown settlement

differentialdeformation

50

70

90

110

130

150

170

190

210

230

250

0 5 10 15 20 25 30 35 40

advance force [MN]

crow

n be

ndin

g m

omen

t [kN

m]

BSM ring 1BSM ring 23D-FEM ring 13D-FEM ring 2

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. 49-56STUVA (editor), 1996, “Eignungsprüfungen 4. Elbröhre Elbtunnel, Reibungsversuche”,

www.stuvatec.de/tubbing_ergebnisse.htmGijsberg, 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

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