ANALYTIC SOLUTIONS OF ELASTIC TUNNELING PROBLEMS Phần 7 ppt

Chapter 7

OVALIZATION OF A CIRCULAR TUNNEL

The application of uneven horizontal and vertical stresses may cause a circular

tunnel with a flexible lining (or no lining) to become oval in shape. Such uneven

stresses are often found in the underground, where the horizontal stresses can

often be expressed as a factor multiplied by the vertical stresses (see Chapter 5).

The ovalization effect due to these uneven stresses can be derived from the

basic Kirsch solution [37] for a stress-free cavity with unequal stresses applied

at infinity, as shown by Pender [25], who deduced the incremental displacements

for this case. The stresses on tunnel linings due to ovalization had already been

discussed by Morgan [21] and Muir Wood [22].

The components of ovalization along the tunnel wall derived from the Kirsch

solution contain a radial component ur = uo cos 2θ and a tangential component

uθ = −uo sin 2θ, where uo is the maximum displacement along the tunnel cav￾ity. This boundary condition, having been derived from the Kirsch solution for

a stress-free cavity, corresponds the case in which zero shear stresses, σrt can

be expected along the tunnel periphery. This corresponds to the case in which

the tunnel lining can be assumed to be frictionless. Uriel and Sagaseta [39]

extended Sagaseta’s original ground loss solution to include these ovalization

components, and obtained surface settlements which are valid for incompress￾ible soils.

Verruijt and Booker [40] extended Uriel and Sagaseta’s work by using an

elastic model for the underground. They obtained expressions for the displace￾ments in the entire field for all values of Poisson’s ratio, and they obtained them

for both the ground loss problem and for an ovalization of a tunnel. For their so￾lution Verruijt and Booker focused on the related mode of ovalization in which

zero tangential displacements are assumed along the tunnel periphery, uθ = 0.

This corresponds to the case in which the tunnel lining is rough relative to the

surrounding ground and little or no sliding occurs between the ground and the

lining.

In this chapter, the ovalization boundary condition used by Verruijt and

Booker is applied to the solution obtained in Chapter 4 in order to obtain the

exact solution for an ovalizing tunnel in an elastic half-plane. These results, first

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