ANALYTIC SOLUTIONS OF ELASTIC TUNNELING PROBLEMS Phần 6 potx

... (r/h<0.01). § 6. 4 Stresses around the Tunnel The isotropic stresses 1 2 (σ xx +σ yy ) and deviatoric stresses  1 4 (σ yy − σ xx ) 2 + σ 2 xy are shown in Figure 6. 6. It appears from Figure 6. 6 that ... for r/h < 0.5 (6. 8) It should be emphasized that this rule of thumb is only valid for a uniform contraction of the tunnel cavity and for linearly elastic behavior. Equation...

Ngày tải lên: 07/08/2014, 04:21

... SOLUTIONS OF ELASTIC TUNNELING PROBLEMS xii Acknowledgements ABSTRACT Analytic Solutions of Elastic Tunneling Problems The complex variable method for solving two dimensional linearly elastic prob- lems ... 41 §5 .6 Comparison to a Finite Element Calculation 44 §5.7 Chapter Summary 46 Chapter 6. The Ground Loss Problem 48 6. 1 Solution of the Problem 48 6. 2 Val...

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... functions which are analytic in R beyond the loaded portion of the surface of the half-plane, including the point at infinity [ 16] (these functions are also analytic in all finite parts of R,as ∗ ϕ(z) ... no longer be analytic. We begin by examining the behavior of the complex potentials for large values of z. Behavior near Infinity The derivatives of ϕ(z)and ψ(z)can be continu...

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... (4.4) The compactness of (4.4) is a result of the cancelation of the logarithmic terms in (3. 26) along the surface, due to symmetry. Since the integration constant along one of the two boundaries ... point on L s directly above the center of the hole. The radius of the hole is denoted by r and the depth of the center of the hole is denoted by h. The geometry of the problem...

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... Tunnel Chapter 4 26 A Deforming Circular Tunnel Chapter 4 The expansion of the right-hand sides of both (4.21) and the modified form of (4.24) is complete. The left-hand sides of these equations ... like powers of σ in the expansion of the modified form of (4.24). This results in a complicated system of equations which are simplified considerably by eliminating c k and d k thro...

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... function of ν. 42 Buoyancy of a Rigid Tunnel Chapter 5 an effective reduction in stiffness of the ground above the tunnel for low values of Poisson’s ratio, as can also be observed in the reduction of ... difference being that a fully analytical solution method has been found for the computation of the coefficients of the complex potentials, as discussed in Chapter 4. § 6. 1 Sol...

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... for other values of ν appear to be within one order of magnitude of the errors given here. 72 Ovalization of a Circular Tunnel Chapter 7 § 7.7 Chapter Summary In this chapter an analytical solution ... in order to obtain the exact solution for an ovalizing tunnel in an elastic half-plane. These results, first 61 66 Ovalization of a Circular Tunnel Chapter 7 -2 -1 0 1 2 -2 -1 0 -2...

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... dimensional linearly elastic problems can be used to obtain several fundamental solutions of tunneling problems. The method was used in Chapter 3 to derive the general mathematical repre- sentation of problems ... studies has been discussed in detail in [ 26, 12]. The ratio of the amount of ovalization to the amount of ground loss is often denoted by ρ. For the exact so...

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... system of equations (4.38) - (4.39) numerically and examining the behavior of p k for large values of k and different values of p 0 , as performed by Ver- ruijt [42]. It should be noted that part of ... use of (B.14). In order to perform these calculations we will need expressions for ∗ ϕ  (z), ∗ ϕ  (z), and ∗ ψ  (z) in terms of ζ . The derivative of ϕ(z) with respect to z...

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... 2α 6 + 4α 4 − 6 2 + 3), e 2 =−α 10 − 4α 6 + 10α 4 − 3α 2 − 2, e 3 = α(2α 8 + 2α 6 − 6 4 − 2α 2 + 4), e 4 = 2α 2 (−α 6 + α 4 + α 2 − 1). Breaking the last expression on the right-hand side of ... smaller zijn dan uit de theorie zou volgen. O.E. Strack, Analytic Solutions of Elastic Tunneling Problems. Ph.D. Thesis, Delft University of Technology. Delft University Pre...

Ngày tải lên: 07/08/2014, 04:21