Reinforcement optimisation around tunnels

Topology optimisation of the reinforcement around underground openings

Figure 3.5 The optimum shape for two intersecting tunnels obtained by Ren et al. (2005).

paper is the external work along the tunnel wall defined as

W (u) = Z

Γ

t · udΓ (3.28)

where u is the displacement vector; Γ is the tunnel’s boundary, and t is the

negating surface traction on tunnel’s boundary prior to excavation. In this

paper the tunnels were assumed to be deep enough so that the difference of

the gravity force is negligible. By assuming linear elasticity, the superposition

principle can be applied. The loading of the tunnel is thus equivalent to the

superposition of two other load cases: the initial in-situ stresses and the

negating surface traction t. Figure 3.6 illustrates this idea.

In their approach Yin et al. (2000) used the homogenisation method to

Figure 3.6 Using superposition principle to analyse a deep tunnel: a) A tunnel under remote stresses; b) the pre-excavation stress state; c) the negating surface traction.

reinforced material consisting a smaller square of size µ < 1 in its centre

made of original rock. The reinforcement material was assumed to be linear

elastic with a Young’s modulus five times that of rock mass.

Yin and Yang (2000a) solved the reinforcement optimisation problem for

tunnels in layered rock structures. This structure may consist of layers of

hard and soft rocks with different Young’s moduli. Reinforcement optimi-

sation of tunnels in four different structures were studied by Yin and Yang

(2000a), namely, isotropic soft, hard/soft, soft/hard, and hard/soft/hard rock

structures.

Yin and Yang (2000a) employed the SIMP method to minimise displace-

ment based objective functions. These objective functions correspond to the

sum of the relative displacements around the opening boundary. In their pa-

per, linear elastic behaviour is assumed for both original and reinforced rock.

The following power-law interpolation scheme is used for stiffness tensor

Eijkl(ρ) = ρpE (r)

ijkl+ (1 − ρ p_)E(o)

where ρ is the relative density; p is the penalty factor, and E(r)ijkl and

E(o)ijkl are stiffness tensors of reinforced rock and original rock respectively.

In their paper, Yin and Yang (2000a) have solved two examples in the

four rock structures considered with deep tunnel assumption (neglecting the

weight of rocks). In another example, however, they included the gravity

force for a tunnel in isotropic media.

The same approach was applied by Yin and Yang (2000b) to find the

optimum reinforcement topology minimising the floor and side wall heaves

of a tunnel in homogeneous rock. In this paper the weight of rock material

was neglected and the tunnel was considered under stress biaxiality.

The reinforcement optimisation of underground tunnels was also studied

by Liu et al. (2008). Different displacement based objective functions were

considered in this paper. To solve the optimisation problem, Liu et al. (2008)

used the BESO method within a fixed grid finite element framework. The

fixed-grid finite element prevents the formation of checkerboard patterns and

smoothens the final topologies. The following interpolation scheme is used

in this paper Eijkl(η) = ηE (r) ijkl+ (1 − η)E (o) ijkl (3.30)

where η is the design variable field changing between 0 and 1. The two ma-

terial phases differ in their Young’s moduli. The sensitivity numbers for the

BESO method can be calculated by using (3.30) in the results of sensitivity

analysis of the objective functions. More information on this issue can be

found in Chapter 4 where the application of the BESO method in solving bi-

Tailoring topology optimisation

algorithm for underground

excavation problems

4.1

Introduction

As mentioned in the previous chapter, in excavation design, the shape of

the opening and the topology of the rock reinforcement can be optimised by

state-of-the-art topology optimisation techniques.

In reinforcement optimisation the material is changing between normal

rock and reinforced rock. The material interpolation scheme is thus different

from solid-void design and the choice of material interpolation scheme is

more critical (Bendsøe and Sigmund 2003). Unlike material-void design, in

bi-material (or multi-material) problems, the ratio of the Young’s moduli of

the two material phases is a finite number. This might lead to convergency

difficulties specially when the elasticity properties of the two materials are

very close to each other.

For optimising the shape of the opening, it is necessary to find the bound-

ary of the opening. The material elements on this boundary may change

to voids and the voids on the inner side of this boundary may change to

material elements. As discussed before, the SIMP and the homogenisation

methods are not very suitable for shape optimisation, because the material-

void boundary is not well definable in these methods. In applying ESO or

BESO methods a normal material-void interpolation scheme would be suf-

ficient provided that the switches between material and void elements are

limited to the elements at the boundary of the opening. This restriction

generally assures that the topology of the opening will not change.

In this chapter a reformulation of the BESO technique is presented. To

derive the sensitivity numbers, a general approach is presented which is based

on sensitivity analysis. The characteristics of the proposed BESO technique

is then tuned and improved to match these special requirements and consid-

erations.