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ANALYTIC SOLUTIONS OF ELASTIC TUNNELING PROBLEMS Phần 3

Có nhiều loại hầm. Loại hầm chui qua sông, eo biển hẹp như đường hầm Seikan, đường hầm eo biển Manche, hầm Thủ Thiêm. Loại hầm chui qua đường dành cho phương tiện cơ giới. Loại hầm chui qua đường dành cho người đi bộ, và đôi khi cả xe đạp. | Section 3.2 Complex Potentials for a Half-Plane with Holes 13 where the notation xb denotes the increase undergone by the expression inside the brackets along the integration path from xa to xb. The integral in 3.18 is path independent in portions of R beyond the expansion circle. Substituting 3.14 and 3.15 in 3.18 gives for values of xa and xb outside a sufficiently large circle centered at the origin and containing all of the holes if tx ity dí T T ln i T - T n Jxa xb I P0b xb xbcp 0 E xb Ỷ0 xb 3.19 - P0 . xa - xap ữí xa - 0b xa - Since 3.19 must remain finite when xa and xb approach infinite values independently the coefficient of the real-valued logarithm must vanish. When xa and xb each approach infinite values simultaneously the integral must approach the sum of all external forces on the half-plane. It follows that we must have T TT 0 and TT - T n Fx iFy 3.20 where total resultant force T T s s h h Fx iFy Fx iFy Fx iFy 3.21 is given by the sum of the resultant forces acting on the surface Fx iFy and on the holes Fx iFy . The corresponding values of T and T agree with the coefficients of the logarithms obtained in 10 for the far-field behavior of the potentials in a semi-infinite plane with holes and vanishing stresses at infinity. Final Form of the Complex Potentials Using 3.20 and 3.16 to calculate Y and Y and substituting the results in 3.8 and 3.9 allows us to obtain the general form of the potentials for a half-plane with holes valid for all values of z in R. Equations 3.1 and 3.2 become p z - s s h h Fx iFy K Fx iFy m - ế k 1 s log z - Zc 2n 1 K k TO y log z - Zk z P0 z 2n 1 K 4 2n k Fx iF1 3.22 14 Multiple Holes in a Half-Plane Chapter 3 and Ỷ z s s h h Fx - iFy Fx - iFy 2n 2n 1 k log z - Zc K FXi l y log z - Zk - - z Ỷ0 z . f 2n 1 k 2 k 1 3.23 We note that in deriving these potentials we assumed that only a finite section of the surface was loaded. This assumption was used to expand the potential functions in Laurent series for large values of z so .