17 Chapter 3 LAMINATED MODEL DERIVATION 3.1 Fundamental Poisson Equation The simplest version of the frictionless laminated model was originally introduced in 1961 by Salamon, and more recently explored in 1991 (Salamon, 1962; Salamon, 1991b; Yang, 1992). In this model, the media consists of a stack of strata laminations where the interfaces between beds, including the ground surface, are all horizontal, and free of shear stresses and cohesion (see Figure 3.1). In the general version of this model, the elastic modulus, Poisson s Ratio, and thickness of the j-th bed are E j, v j, and t j, respectively. The theory of thin plates is used as the basis of the model (Ugural and Fenster, 1975). From this theory, the relationship between the vertical deflection (w) of Figure 3.1 Schematic of laminated overburden.
18 the middle plane of a horizontal plate and the resultant transverse pressure (p) acting on the plate is defined by: (3.1) where D is the flexural rigidity of a plate: (3.2) and 4 denotes the bi-harmonic operator in the xy plane, specifically: (3.3) Throughout this thesis, positive normal stresses and strains signify compression, and the positive z-axis points vertically downward. This sign convention necessitates that vertical displacement in the upward direction is taken to be positive. As shown in Figure 3.1, the resultant transverse pressure on the (j+1)-th plate can be written as: (3.4) where b and t denote the induced vertical stress at the bottom and the top of the (j+1)-th plate, respectively. It should be noted that in the first approximation, the thickness (t) of the plate will not change provided that no traction acts on the face of the bent plate and that the stretching of the middle plane is neglected (Ugural and Fenster, 1975). Thus, the change in thickness of an individual plate can be attributed solely to the effect of the vertical stress.
19 If we assume that the compressive stress acting across the top half of layer j+1 and the bottom half of layer j is t, then the change in distance between the middle planes of the two layers is: (3.5) Solving for the stress on the top of the (j+1)-th layer: (3.6) Similarly, the stress on the bottom of the (j+1)-th layer is: (3.7) Substituting equations 3.6 and 3.7 into equation 3.4 gives: (3.8) Factoring out common terms: (3.9)
20 Adding a few additional terms in preparation for a finite-difference representation: (3.10) and rearranging: (3.11) If we use to represent the finite-difference operator such that: (3.12) Then, the previous equation 3.11 can be written as: (3.13) This equation controls the strata pressure for a frictionless laminated model containing distinct layers. If we consider the simplification where the overburden is composed of
21 homogeneous stratifications such that the thickness (t), elastic modulus (E) and Poisson s Ratio (v) are identical for each layer, then the above equation (3.13) simplifies greatly to: (3.14) Substituting back into equation 3.1 we get: (3.15) Now, if the thickness of the layers are small in respect to the areal extent of the problem, the finite-difference approximation in equation 3.15 can be represented as a differential operator: (3.16) Substituting this differential operator into 3.15, simplifying and rearranging gives: (3.17) Making a substitution of: (3.18)
22 we get: (3.19) This is a fourth-order, partial-differential equation, which mathematically transforms the homogeneous, layered medium into a quasi-continuum that maintains the flexibility of the laminated rock mass. In order to eliminate the bi-harmonic operator ( 4 ) and solve equation 3.19, a double Fourier integral transform is used. The Fourier transform of the vertical displacement, w(x,y,z), with respect to x and y is defined as (Weinberger, 1965): (3.20) where w is the transformed vertical displacement function and 1 and 2 are the transformation variables. In conjunction with the Fourier transform, the inverse Fourier transform is defined as: (3.21) The feature of the Fourier integral transform that makes it useful for solving higher order differential equations and for reducing partial differential equations to ordinary differential equations is that the transform of the derivative of the original function is simply the transform of the original equation multiplied by -i. In our case, this feature can be expressed as: (3.22)
23 If the double Fourier transform is applied to equation 3.19, the result is the transformed equation: (3.23) where: (3.24) The general solution for equation 3.23 takes the form: (3.25) where C 1 and C 2 are the constants of integration to be determined from boundary conditions. To solve this equation (3.25), an infinite laminated medium is assumed, and the origin of the z-axis is set to be at the seam level with the positive z-axis pointing downward. Also, compression on the seam is specified as positive and upward movement of the overburden is specified as positive. Next, the domain of the vertical displacement is broken into two parts; above the seam where z is negative (w (-) ) and below the seam where z is positive (w (+) ). Then, applying the boundary condition that the vertical displacement must go to 0 (w 0) as the distance from the seam goes to infinity (z ± ), the solution for the displacements in the roof and floor takes the form: (3.26) where symmetry between the roof and floor displacements dictates that the constant of integration, C, is the same for both equations. Also, knowing that the vertical stress,, is related to the change in displacement by the elastic modulus, E;
24 (3.27) the equations for the vertical stress can be written as: (3.28) By examination of equation 3.26, it can be seen that: (3.29) Rearranging these equations we get: (3.30) Now, using the critical identity that the inverse Fourier transform of: (3.31) where 2 is the Laplace operator in the x-y plane: (3.32)
25 the inverse transform can be taken of equations 3.30: (3.33) (3.34) Then, since at the seam level the stresses on the roof and floor are equal: and the convergence, s, can be represented as: (3.35) the equations 3.33 can be rearranged as: (3.36) Now expanding the Laplace operator and realizing that the seam level stresses are induced stresses, the fundamental equation for a laminated overburden with homogeneous stratifications can be written: (3.37) This second-order, partial-differential equation relates the convergence in the seam to the induced stress at the seam level in a layered media.
26 3.2 Displacement and Stress Influence Functions Equation 3.37 can be used to solve for the displacements at the seam level once the lamination properties are determined. However, in order to solve for displacements and/or stresses remote from the seam, an influence or kernel function which relates the seam convergence to the remote displacement needs to be derived. Following Yang s (1992) derivation, the boundary conditions for a concentrated unit convergence,, applied at (0, 0, 0) are: (3.38) Then the Fourier transform of the first boundary condition is: (3.39) Taking equations 3.26 at z = 0, and substituting these into equation 3.39, the value of C can be found as: (3.40) Substituting this result back into equations 3.26 and then taking the inverse Fourier transform results in the influence function for vertical displacement (W) from a unit point seam convergence: (3.41)
27 Using the identity in equation 3.27, the above equation can be used to determine the influence function for vertical stress from a unit point seam convergence: (3.42) 3.3 Numerical Solution of Fundamental Equation The fundamental differential equation which relates the convergence (s) in the seam to the induced stress ( i ) at the seam level in a layered media was derived in equation 3.37 and is repeated here: (3.43) This is a classic second-order, elliptical, partial-differential equation, and many numerical techniques have been developed for solving this type of equation. Examining the right side of this equation, it is found that the induced stress is the only variable which is not a material constant. In the simplest case of an opening in the seam, the induced stress ( i ) is equal to the negative of the primitive ( q ), or overburden stress. However, when there is material in the seam supporting the roof, the induced stress in the surrounding laminations is reduced by the support of the coal or other seam material ( c ). In general, the amount of support provided by the seam material is a function of the seam convergence, c (s), and in the case of failed material or gob, this support would typically be a non-linear function of the seam convergence. The calculation of surface-effect stresses ( s ) are more complicated. For calculating the effects of a traction-free plane at the ground surface, the technique of a mirror-image seam is used (Yang, 1992; Salamon 1991b). Initially, the seam is considered to
28 be in an infinite medium and the appropriate seam displacements are determined. Then, a fictitious mirror-image seam is placed above the ground surface at a distance equal to the depth (see Figure 3.2). This fictitious seam is also assumed to be in an infinite medium; however, the calculated convergence in the actual seam is exactly mirrored as divergence in the mirror-image seam. Thus, the distributions of convergence and divergence are identical in magnitude but opposite in sign. Consequently, the sum of the propagated displacements and stresses (after equations 3.41 and 3.42) from the two seams is zero at a plane midway between the two seams, at the ground surface. Thus, the union of the two infinite media solutions corresponds to the effect of the actual seam at finite depth. Figure 3.2 Schematic of mirror-image and multiple-seam stress calculation.
29 However, at the level of the actual seam, the propagated stresses from the mirror-image seam contribute to the total induced stress on the actual seam elements. In fact, every element in the mirror-image seam propagates a small incremental stress to every element in the actual seam based on equation 3.42. So, the surface effect stress ( s ) on any given seam element is equal to the numerical integration of the incremental stresses propagated from the mirror-image seam. (The details of the surface-effect stress calculation are explained further in section 5.5.2.) The calculation of multiple-seam stresses ( m ) is very similar to the calculation of surface-effect stresses. However, instead of a mirror-image seam with mirrored divergence, for the multiple-seam stress calculation, the second seam has an independent mine plan and therefore, an independent displacement distribution (see Figure 3.2). Once again, every element in the second seam propagates a small incremental stress to every element in the actual seam (and vice versa). Therefore, the multiple-seam stress ( m ) on any given seam element in the actual seam is the numerical sum of the incremental stresses propagated from every element of the second seam. (The details of the multipleseam stress calculation are explained further in section 5.5.1.). From the proceeding paragraphs, it is seen that the surface-effect and multiple-seam components of the induced stress are extremely complicated functions of in-seam and off-seam convergence. Thus, in general, the total induced stress can be the sum of many factors, some of which may be non-linear functions of the seam convergence. (3.44) Once it is determined that the induced stress can be a non-linear function of the seam convergence, the choice of solution techniques is generally limited to iterative procedures. Also, from a practical perspective, a robust solution algorithm with accelerated convergence is desirable, and to stay compatible with MULSIM it is desired to solve the seam convergence distribution on an even grid. Considering these factors, a
30 central-difference approximation using a Gauss-Siedel iterative scheme with Successive Over-Relaxation (SOR) was chosen as the solution technique (Ames, 1992). With this numerical technique, equation 3.43 can be solved over a gridded area using successive iterations of the kernel equation: (3.45) where: O is the over-relaxation factor (between 1 and 2); x is the grid dimension; the superscript r refers to the iteration number; and the subscripts j and k refer to the horizontal and vertical grid locations, respectively, on the finite-difference grid such that s 2,2 is the convergence value at the grid intersection 2 over and 2 up from the origin. This Gauss-Siedel solution scheme with SOR provides a number of numerical and computational advantages for the practical solution of equation 3.43. Adjustment of the over-relaxation factor (O) allows the number of iterations for the convergence of the finite-difference solution to be minimized. Using the convergence values that were recently updated during the present iteration improves the speed of convergence and allows the convergence values to be stored in a single array that is updated as the solution progresses through the grid. Finally, the iterative solution technique allows the calculation of the non-linear induced stress to be smoothly incorporated into the normal iteration cycle.