Stability Problems In Tunnelling

20 Stability problems in tunnelling

Leaving aside a precise definition of mechanical stability we only need to mention that a loss of stability occurs if in the course of a loading process a mechanical system becomes suddenly (often the term ’spontaneously’ is used) softer, so that large deformations appear. These can cause serious damage. The most widespread known stability problem is the buckling of a rod. Here we consider the buckling of a tunnel lining or, equivalently, a pipe.

20.1 Rockburst Rockburst can occur in deep tunnelling and mining and is manifested by spalling of the cavity walls. Large amounts of stored elastic energy are released and transformed into kinetic energy so that rock plates of several dm thickness are accelerated into the cavity causing worldwide many casualties per year. The mechanism of rockburst is not yet completely understood. The thickness df of the spalled rock can be estimated according to the following empirical formula:1 σmax df = 1.25 − 0.51 ± 0.1 (20.1) r qu where r is the tunnel radius, σmax is the maximum circumferential stress at the tunnel wall and qu is the unconfined strength of the rock.

20.2 Buckling of buried pipes The buckling under consideration is caused by external forces acting upon the pipe (or tunnel lining). We should distinguish between forces exerted by a fluid and forces exerted by the ground. The latter depend on the deformation 1 P.K. Kaiser et al., Underground works in hard rock tunnelling and mining. GeoEng 2000, Melbourne

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20 Stability problems in tunnelling

of the pipe, whereas fluid loads are independent of deformation and always normal to the pipe surface. 20.2.1 Buckling of pipes loaded by fluid We consider the pipe as a beam with initial curvature, where the differential equations of bending can be applied. Plane deformation (i.e. displacements ur = 0, uθ = 0, uz ≡ 0) implies that the Young’s modulus E usually applied in bending theory of beams has to be replaced by E ∗ := E/(1 − ν 2 ). The differential equation follows from the known relation ΔM = EJ · Δκ, where Δκ is the change of the beam curvature and J is the moment of inertia of the area shown in Fig. 20.3. With r := r0 + u, r˙ := dr/dθ = r0 dr/ds and r := dr/ds we can express the curvature as κ=

r2 + 2r˙ 2 − r¨ r 2 2 3/2 (r + r˙ )

Introducing r = r0 + u and neglecting terms quadratic in u and u as well as the product uu we obtain: κ≈

1 u − 2 − u r0 r0

.

With κ0 = 1/r0 for the initially prevailing circular form we finally obtain Δκ = κ − κ0 ≈ −

u − u r02

,

and the differential equation u +

u M =− ∗ 2 r E J

(20.2)

or u ¨ + u = −r2

M . E∗J

(20.3)

We now assume that the buckled shape of the pipe is symmetric with respect to the x and y axes, i.e. u(ϑ) = u(−ϑ)  π  −ϑ = u +ϑ u 2 2 π

(20.4) (20.5)

Referring to Fig. 20.2 we can express the bending moment M in dependence of ϑ: M (ϑ) = M0 − N0 [r0 + u0 − (r0 + u) cos ϑ] 1 1 2 − p [r0 + u0 − (r0 + u) cos ϑ] − p(r0 + u)2 sin2 ϑ . (20.6) 2 2

20.2 Buckling of buried pipes

353

Fig. 20.1. Buckled pipe

Fig. 20.2. Deriving the bending moment as function of ϑ

Taking into account that N0 = p(r0 + u0 ) and neglecting small terms2 yields from (20.6): M ≈ M0 − pr0 (u0 − u) .

(20.7)

Introducing (20.7) into (20.3) yields ..

u +k 2 u = −

r02 (M0 − pr0 u0 ) E∗J

(20.8)

with k 2 := 1 +

r02 p E∗J

.

(20.9)

The solution of (20.8) reads u = A cos kϑ + B sin kϑ −

(M0 − pr0 u0 )r02 E ∗ Jk 2

From (20.4) follows B = 0 and from (20.5) follow 2

i.e. terms of the order of u2

.

(20.10)

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20 Stability problems in tunnelling

  π   π cos k( − ϑ) = cos k( + ϑ) 2 2 or

 π sin k sin(kϑ) = 0 2

,

 π sin k =0 . 2

It follows k = 2n with n = 1, 2, 3, . . . . Thus, the smallest (critical) buckling load pcr is obtained (Grashof, 1859) from (20.9) for k = 2 as pcr = 3

E∗J r03

.

20.2.2 Buckling of elastically embedded pipes We consider a pipe embedded within a material in such a way that the interaction is governed by the subgrade modulus Kr . According to Nicolai3 , the buckling load is obtained as p = (k 2 − 1)

r E∗I + Kr 2 r3 k −1

,

(20.11)

with k = 2, 3, 4, . . .. If we introduce into Equ. 20.11 the number k which minimises p 4 , we obtain the critical buckling load according to Domke and Timoshenko: 2 pcr = Kr E ∗ J . (20.12) r

Fig. 20.3. Area referring to moment of inertia J

3 E.L. Nicolai, Stabilit¨ atsprobleme der Elastizit¨ atstheorie, Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 3, 1923, 227-229 dp 4 This can be obtained by formal differentiation, dk = 0.