ANALYTIC SOLUTIONS OF ELASTIC TUNNELING PROBLEMS Phần 2 ppsx

Part I INTRODUCTION AND THEORY 1 Chapter 1 INTRODUCTION Tunneling in urban environments requires the prediction of surface movements in order to asses possible effects on nearby structures. Currently, the most commonly used means of making these predictions has been the application of empirical formulas such as the one by Peck [24]. Several analytic models have also been proposed for this purpose [28, 30, 40], which are based (at least in part) on linearly elastic models of soil and which include the effect of the free soil surface in an approximate fashion. Sagaseta [28] showed that an incompressible linearly elastic model is appropriate for describing undrained soil behavior. Elastic models, however, may also be appropriate for drained cases if the stresses remain such that soil plasticity is limited. Another issue facing the designers of tunnels is the determination of the strength of the tunnel lining, based on the stress induced on the lining by the surrounding material. For tunnels excavated in rock most of these calculations are based on linearly elastic solutions for infinitely deep circular tunnels [37], elasto-plastic solutions for infinitely deep circular tunnels [11, 7], or variations of these models [45, 4]. Solutions for shallow tunnels, modeled as stressed or stress-free circular holes in elastic half-planes, were obtained by Jefferey [15] and Mindlin [18, 19], respectively, and have also been used for this purpose. The stresses on tunnels excavated in soil are often determined by means of spring models representing tunnel-soil interactions [9] or by empirical approaches such as presented by Peck [24]. Both of the above issues will be treated in this thesis through means of analytic, linearly-elastic solutions for shallow circular tunnels. These two- dimensional solutions fully take into account the effects of the stress-free surface and the so-called buoyancy effects, which arise as a result of the unloading of the material under the tunnel during excavation. Both effects are key elements in accurately determining the stresses acting on the tunnel and in determining the formation of the settlement trough along the surface. In addition to their possible use as prediction tools, the solutions presented here can serve as par- tial validations of numerical calculations based on more complex constitutive models of soil. 3 4 Introduction Chapter 1 The solutions are obtained through use of the complex variable method, developed by Muskhelishvili and described in [23, 34, 10]. Muskhelishvili himself used the method to solve the problem of a stress-free elliptical opening in an infinite plane [23]. Yu [45] used the method to solve the problem of a tunnel in an infinite gravity-loaded medium that includes the buoyancy effect, but disregards the effect of the free surface. The complex variable method was also used by Sherman [33], who obtained a system of integral equations for an elliptical tunnel reducing to a slit in a half-plane. Verruijt recently used the method to find a solution for a deforming circular tunnel in a half-plane [42]. The solution consists of two infinite series whose coefficients are determined in part by a numeric limiting process. In this thesisVerruijt’s solution is extended to include buoyancy effects, and is reworked in order to determine all coefficients analytically. This thesis is divided into two parts: the first part (Chapter 1 – Chapter 4) deals with the mathematical theory involved in determining the analytical solu- tions of tunneling problems using the complex variable method, and the second part (Chapter 5 – Chapter 9) deals with applications of the solutions and presents the conclusions. Part one begins with this introduction (Chapter 1) and a brief explanation of the complex variable method (Chapter 2). The method is then extended for the solution of problems with resultant forces acting on multiple holes in a half- plane in Chapter 3. Finally, the solution for a deforming circular tunnel in an elastic half-plane that includes buoyancy effects is derived in Chapter 4. Part two begins with the treatment of tunnel deformations in three separate chapters: the buoyancy of a rigid tunnel (Chapter 5), the ground loss around a tunnel (Chapter 6), and the ovalization of a tunnel (Chapter 7). In Chapter 8, these deformation modes are combined in order to illustrate their possible use for the prediction of settlement troughs. Finally, Chapter 9 sums up the primary conclusions that can be drawn from the research presented in the thesis. Chapter 2 THE COMPLEX VARIABLE METHOD The complex variable method for the solution of two-dimensional linearly- elastic problems [23, 34, 10] involves the use of two analytic functions of a complex variable. Complex variables define points in a two-dimensional xy-plane in the form z = x + iy, where x is the real part of z (also written {z}), and y is the imaginary part of z (also written {z}). The imaginary constant is denoted by i, and the complex conjugate of the variable z is written as z = x − iy, which is obtained by replacing all occurrences of i by −i. A complex function is analytic at a point z 0 if it is continuous and has the same derivative along every possible approach to z 0 . The properties of analytic functions mentioned above lead to many conse- quences, one of which is that the real and imaginary parts of analytic functions are harmonic, satisfying Laplace’s equation. It is primarily this property which is used to prove that the horizontal and vertical displacements, u and v,inan elastic region R can be expressed as 2µ(u + iv) = κϕ(z) − z ϕ  (z) − ψ(z), (2.1) where the functions ϕ(z) and ψ(z) (also referred to as the complex potentials) of the complex variable z are analytic in R and where the notation ϕ  (z) denotes differentiation with respect to the variable enclosed in parenthesis (z in this case). The elastic constant µ is the shear modulus of the elastic material, and the constant κ is related to Poisson’s ratio ν by κ = 3 − 4ν (2.2) for plane strain, and by κ = 3 − ν 1 + ν (2.3) for plane stress. The stresses can be expressed using the same potentials as σ xx + σ yy = 2 {ϕ  (z) + ϕ  (z)}, (2.4) σ yy − σ xx + 2iσ xy = 2 {zϕ  (z) + ψ  (z)}, (2.5) where σ xx and σ yy are the normal stresses in the x and y directions, respectively, and σ xy is the shear stress. Note that only the derivatives of the potentials ϕ(z) 5 6 The Complex Variable Method Chapter 2 and ψ(z) are present in (2.4) and (2.5), while the potentials themselves appear in (2.1). Two-dimensional elasticity problems are solved using the complex variable method by determining ϕ(z) and ψ(z) from given boundary conditions. Equa- tion (2.1) can be used for this purpose along boundaries where the displacements are given. Along boundaries where the stresses are specified it is possible to use (2.4) and (2.5) or transformations of these equations. However, it is usu- ally more convenient to use the following equation, which relates the integral of the horizontal and vertical tractions t x and t y along a curve to the complex potentials: i  z z 0 (t x + it y ) ds = f(z)= ϕ(z) + zϕ  (z) + ψ(z) + C z 0 , (2.6) where s is a coordinate on the curve and the part of the region on which the tractions are acting lies to the left as the curve is traversed from z 0 to z. The constant of integration C z 0 is a result of the arbitrary starting value z 0 of the integral. The loading function f(z)is the total force exerted on the curve from z 0 to z, multiplied by the imaginary constant i. It should be emphasized that either the stresses or the displacements can be given along a boundary, but not both, as this assumes a prior knowledge of the solution of the problem. Other formulations, in which certain components of the stresses are given together with certain components of the displacements (sometimes referred to as mixed boundary conditions), are not treated in this thesis. One of the advantages of the complex variable method in solving elasticity problems is that both the stresses and displacements are solved for simultane- ously, and that the boundary conditions for the displacements and stresses are similar. This allows for similar techniques to be used in the solution of both boundary-value problems. Another advantage is that the many theorems that follow from the theory of analytic functions, such as Cauchy’s Integral Theo- rem, Laurent’s Theorem, and theorems on conformal mapping, can be used to generate elegant solutions for difficult problems. Chapter 3 MULTIPLE HOLES IN A HALF-PLANE Two dimensional elasto-static problems in half-plane regions are greatly compli- cated by the presence of the surface of the half-plane, along which the boundary conditions must extend to infinity. An extension to the complex variable method for half-planes containing holes is the focus of this chapter. When holes, inhomogeneities, or cracks are present in the half-plane, the problem is often solved by superimposing the reflections of solutions for an infinite plane across the boundary of the half plane. For examples of half-plane problems using the complex variable method see [20, 5, 14, 27, 38]. When problems form equilibrium systems within the half-plane it has been shown that direct approaches, not based on reflections across the half-plane, can be used to obtain solutions [33, 42, 43]. If the desired solutions do not form equilibrium systems within the half-plane, it is more difficult to solve the problem without resorting to superposition. Such cases may arise, for example, for the excavations of tunnels in gravity loaded mediums, in which buoyancy forcesarise to the removal of the weighted material [18, 41]. Direct application of the potentials first derived in [23] for an infinite plane containing holes with resultant forces acting on them leads to difficulties in meeting the boundary conditions along the surface. This is shown in [10] and in this chapter. The representation of the potentials for loaded holes in a half-plane has been derived in [10], but only for points in the half-plane far from the region of interest, and with vanishing stresses at infinity. It is the purpose of this chapter to derive the general form of the complex potentials for a half-plane with holes, valid in the entire region and including non-zero stresses at infinity. The methods and notation used closely follow [23], §90, in which the complex potentials for a loaded half-plane without holes are derived. § 3.1 Complex Potentials for a Finite Plane with Holes The complex potentials in a linearly-elastic region R b bounded externally by a finite closed contour (curve) L s and containing m holes bounded by m closed 7 8 Multiple Holes in a Half-Plane Chapter 3 contours (curves) L 1 ,L 2 , , L m can be shown (see [23] and Appendix A) to have the form ϕ(z) =− m  k=1 k F x + i k F y 2π(1 + κ) log(z − z k ) + ∗ ϕ(z), (3.1) ψ(z) = m  k=1 κ( k F x − i k F y ) 2π(1 + κ) log(z − z k ) + ∗ ψ(z), (3.2) where k F x +i k F y is the resultant force on the contour L k , which is assumed known beforehand. The points z k are located arbitrarily within the contours L k , and log(z − z k ) is the multi-valued natural logarithm (with its primary singularity located at z k ). The functions ∗ ϕ(z) and ∗ ψ(z) are single-valued and analytic in R b . It should be noted that due to the presence of the logarithmic terms, the integral of the tractions along a curve (2.6) is not path independent in multiply connected parts of the region. § 3.2 Complex Potentials for a Half-Plane with Holes We are interested in the complex potentials for a lower half-plane containing m holes. We denote the region consisting of the lower half-plane minus m holes as R and note that R is not bounded. The general shape of R and the chosen coordinate system are shown in Figure 3.1. In this case the contour L s in (3.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y L s L 1 L 2 L 3 L 4 L 5 R Figure 3.1: Half-plane with holes. and (3.2) becomes the infinite surface of the half-plane, and R extends to and Section 3.2 Complex Potentials for a Half-Plane with Holes 9 includes the point at infinity, where the functions ∗ ϕ(z) and ∗ ψ(z) may 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 continued analytically through unloaded portions of the half-plane [23, 10]. If we restrict ourselves to cases in which only a finite section of the surface of the half-plane is loaded, then the derivatives of the potentials (which represent the stresses in (2.4) and (2.5)) can be continued analytically near infinity and can therefore be represented by Laurent series (see [23] and [16]) for values of z beyond the loaded portion of the surface. It follows that we can write: ∗ ϕ  (z) = ∞  −∞ a n z n and ∗ ψ  (z) = ∞  −∞ a  n z n . (3.3) If we further limit ourselves to cases in which the stresses remain finite through- out R (and restrict the total load on the surface of the half-plane to finite values), then all exponents of z which are greater than zero in (3.3) must vanish in light of (2.4) and (2.5). It follows from integration that, for large values of z, we can write ∗ ϕ(z) = γ log z +z + ∞  n=0 b n z −n , (3.4) ∗ ψ(z) = γ  log z +   z + ∞  n=0 b  n z −n , (3.5) where the constants γ and γ  denote the coefficients of 1/z in (3.3), where  = B + iC and   = B  + iC  (3.6) denote the coefficients a 0 and a  0 , and where b n and b  n denote the rest of the integrated terms of the Laurent expansions. The first two terms in (3.4) and (3.5) are singular at infinity, whereas the summations in these equations correspond to 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) and ∗ ψ(z) are required to be analytic there). Components of the Potentials in the Lower Half-Plane The logarithms appearing in (3.4) and (3.5) are necessary in order to limit the magnitude of the surface tractions near infinity, as can be seen by choosing γ = γ  = 0 and substituting (3.1) and (3.2), together with (3.4) and (3.5), 10 Multiple Holes in a Half-Plane Chapter 3 in (2.6). These logarithms, written for large z, may actually represent multiple logarithms for smaller values of z in the near-field, as can be seen by considering expansions of the type (see [13]) log(z − z 0 ) = log z + log(1 − z 0 z ) = log z − ∞  n=1 1 n  z 0 z  n , (3.7) where z 0 is an arbitrary point in the complex plane. Any logarithms in the near- field can be combined for large z through use of this expansion. The summation in (3.7) corresponds to the Laurent series of a single-valued analytic function which vanishes at infinity. The singularities of the multiple logarithms which may be part of ∗ ϕ(z) and ∗ ψ(z) cannot be located anywhere within R, as that would violate the analytic nature of the potentials there. In addition, none of the singularities can be located within the holes, as that would violate the expressions for the resultant forces acting on the holes. The conclusion is that any logarithmic singularities contained in ∗ ϕ(z) and ∗ ψ(z) must be located in the upper half-plane. Since logarithms in the upper half-plane can be combined in the lower half-plane using variations and translations of (3.7), the logarithmic components in each of the functions ∗ ϕ(z) and ∗ ψ(z) can be represented by a single logarithm with a singularity in the upper half-plane, plus a single-valued analytic function valid everywhere in R and vanishing at infinity. It follows that we can write ∗ ϕ(z) = γ log(z − z c ) + z + ϕ 0 (z), (3.8) ∗ ψ(z) = γ  log(z − z c ) +   z + ψ 0 (z) (3.9) for all values of z, where ϕ 0 (z) and ψ 0 (z) are single-valued analytic functions in R, including the point at infinity, and where we have chosen, for convenience, the conjugate of some point z c in the lower half-plane (for example, at the center of one of the holes) as the location of the primary singularity of the logarithms in the upper half-plane. It should be noted that the special case in which the singularities of the log- arithms contained in ∗ ϕ(z) and ∗ ψ(z) are located on the upper boundary of the lower half-plane has been excluded in the analysis above. Although theoreti- cally any singularities on the boundary of the lower half-plane would violate the analytic nature of the potentials there, Muskhelishvili [23] has shown that such cases actually represent concentrated forces, if the boundary is altered to contain an infinitesimal semicircle around the singularity. The analysis above can be extended to include the case mentioned here through superposition of the corresponding potentials and a proper calculation of the coefficients γ and γ  of the logarithms in (3.8) and (3.9) (see the analysis below). Such an extension is omitted here. [...]... determined in part by considering the contribution of and to the integral of the surface tractions, (2. 6): z+ z+ z = (2B + B − iC )x − [C + i(B − 2B)]y (3.10) For large values of z along the surface the first term becomes infinite, which violates the condition that the total load acting on the surface is bounded This singularity can be removed by setting 2B + B = 0 C = 0 and (3.11) We assume that the stresses... determined by substituting (3.8), (3.9), and (3.11) into the stress equations (2. 4) and (2. 5) and taking the limit as z approaches infinity along the surface This yields B= ∞ σ xx and 4 B =− ∞ σ xx 2 (3. 12) The constants and have now been determined It is interesting to note that it follows from (2. 4) and (2. 5) with (3.11) and (3. 12) that we must have ∞ σ yy ∞ = σ xy = 0 (3.13) in order for the problem to... Holes 11 Section 3 .2 Determination of the Constants Governing the Stresses at Infinity The constants and in (3.8) and (3.9) can be determined by assuming that the stress distribution at infinity is known beforehand The components B, B , and C of and , given in (3.6), govern the behavior of the stresses at infinity The component C of determines the amount rigid body rotation at infinity (see [23 , 34, 10]) We... resultant forces acting on infinite portions of the half-plane In order to simplify the calculations, we expand and combine the logarithms in the potentials (3.1) and (3 .2) together with the logarithms in (3.8) and (3.9) through use of (3.7) We combine the series resulting from the use of (3.7) with the analytic functions ϕ0 (z) and ψ0 (z) to form two new analytic functions ϕ0 (z) and ψ0 (z), which... are single valued and vanish at infinity Thus we have, for 12 Multiple Holes in a Half-Plane Chapter 3 values of z outside a circle centered at the origin and containing all of the holes, ϕ(z) = ϒ log z + ψ(z) = ϒ log z − ∞ σ xx 4 ∞ σ xx 2 z + ϕ0 (z), (3.14) z + ψ0 (z), (3.15) where h h h F x + i Fy ϒ =γ − 2 (1 + κ) and h κ(Fx − iFy ) ϒ =γ + , 2 (1 + κ) (3.16) in which the total resultant force acting... resultant force acting on the holes is given by h h m Fx + i Fy = k k Fx + i F y (3.17) k=1 The load acting on the portion of the half-plane beyond the expansion circle must equal the sum of the external forces on the holes and on the corresponding section of the surface (see Figure 3 .2) This load can be calculated by integrating the tractions along a curve outside the expansion circle from a point za... expected, that an infinite resultant force on the surface of the half-plane would be needed in order to counteract a constant vertical stress or a constant shear stress at infinity This case has been excluded from the present analysis, but can be obtained by superimposing constant vertical and shear stress fields Determination of the Coefficients of the Logarithms The unknown coefficients γ and γ in (3.8)... tractions along a curve outside the expansion circle from a point za = xa on the positive x-axis to a point zb = xb on the negative x-axis (which keeps the region to the left as stipulated for (2. 6), see Figure 3 .2) y ... zc R Figure 3 .2: Resultant forces acting on an integration path The integral is given by xb i xa (tx + ity ) ds = ϕ(z) + zϕ (z) + ψ(z) xb xa , (3.18) . METHOD The complex variable method for the solution of two-dimensional linearly- elastic problems [23 , 34, 10] involves the use of two analytic functions of a complex variable. Complex variables define. derivatives of the potentials ϕ(z) 5 6 The Complex Variable Method Chapter 2 and ψ(z) are present in (2. 4) and (2. 5), while the potentials themselves appear in (2. 1). Two-dimensional elasticity problems. half-plane, the problem is often solved by superimposing the reflections of solutions for an infinite plane across the boundary of the half plane. For examples of half-plane problems using the complex