Analysis of disturbing influence of traffic load on soil body Janat Musayev,1 Algazy Zhauyt,2 1 Department of Transport engineering and technologies, Kazakh Academy of Transport and Communications named after M.Tynyshpayev, Almaty 050012, Kazakhstan 2 Department of Applied Mechanics and Basics of Machine Design, Kazakh National Technical University named after K.I.Satpaev, Almaty 050013, Kazakhstan Correspondence should be addressed to Janat Musayev; mussaev1975@mail.ru Stress waves propagate in soil in case of earthquake and man-made effects (traffic flow, buried explosions, shield-driven pipes and tunnels, etc.). The wave point-sources are those located at the distances equal to more than two-three waves lengths, that significantly simplifies solving of a problem of these waves strength evaluation. Distribution of stress and displacement by the stress waves propagation in elastic medium is a complex pattern. The stress distribution in propagating waves depends on a type and form of source, conditions of the source contact with medium and properties of mediums in the vicinity of the source. The point-sources and their combinations are selected in such a way to model an influence of machines and processes on soil body in case of shield-driven pipes (tunnels). Key words: waves * propagation * equations * concentrated * tunnels 1. Introduction In confined environment the wave pattern becomes more difficult due to the boundary reflection of waves. For evaluation of different factors influence on the nature of waves propagation it is convenient to divide the waves propagation problem into several stages. At the first stage tasks of waves propagation in infinite medium from the point-sources of the following different types are reviewed: - concentrated force, - double force without moment (two concentrated forces acting contrariwise along one line and applied at small distance 2h from each other), - two double forces without moment (double couple without moment) acting at right angles and in a single plane, - three double forces without moment acting in three orthogonally related directions (center of expansion), - double forces with moment (couple of forces), - combination of two couples of forces with sum of moments equal to zero, - uniform pressures in alveole at a section in length d, - tangential stresses applied at a section of alveole contour in length d, - tangential stresses applied to alveole contour in length d. Solution of tasks enables evaluation of types of wave sources influence on a type and parameters of waves propagating over a distance. At the second stage tasks of wave propagation from the point-sources in semi-infinite elastic medium are reviewed. At this stage tasks of vibration of semi-infinite elastic medium surface from different point-sources of waves at various depths with account of waves reflected from free surface are solved. The solutions make it possible to evaluate a dynamic impact on the environment in case of earthquake, as well as the dynamic impact of different devices and machines used for underground works. This article deals with a task of waves propagation from the point-sources of different types in the infinite space. In the performance of the tasks the Fourier integral transform and generalized functions are used[1,3,4]. 2. Materials and Methods For dynamic tasks both infinite and semi-infinite spaces and for bounded areas we shall use differential equations of motion U j ,ii U i ,ij U j f j , i, j 1,2,3 . (1) Supposing that in the generalized functions outside area occupied by the elastic area the displacement and stress functions are equal to zero[1]. Then equation (1) may be in a form: U j ,ii U i ,ij U j U j cosn, xi s ,i U i cosn, xi s , j ji s cosn, xi s U k t 0 t U t T j t T t U U k t 0 k t T t T , (2) ij - Kronecker symbol, s - delta function at the area boundary, t , t - delta function and its Where U time derivative. ij j s and ij s and - jump of U j functions upon outside passage through the boundary of area . Since outside of this area these functions are equal to zero, symbols U j and ij 2 12 22 32 wave velocity present values of these functions at the area boundary. U , U , U , U Functions j t 0 j t T j t 0 j t T represent initial and final conditions, i.e. displacements and velocities of medium points at t 0 , t T . In the expressions below and above the summation over repeated indices is performed. Let us introduce the notations: X j U j cosn, xi s ,i U i cosn, xi s , j k t T j (3) interval (0, Т), and is equal to zero outside of this area and interval. Since all functions represented in the finite functions are equal to zero outside the area, consequently jumps of functions at the area boundary are boundary conditions. Depending on the assigned tasks a part of these functions is set, the other is defined in the course of solution[11]. Let apply the Fourier transform for the differential equation system solution. Let us multiply the left and right expi k x k and integrate with respect of four variables: x1 , x 2 , x3 , , in member of the equation by other words let apply the Fourier transform to system of equations (2). Here 1 , 2 , 3 means parameters of the - frequency. For description of the Fourier functions let in symbol the transform are Fourier ij ij ,U j , F j ,U j , F j . The following properties of used upon U i ~ i j U i or in tonsorial x j ~ ~ U i , j i j U i and Ui 2 U i . - ratio of P- to S-wave velocity . Having used the agreed notations and the Fourier transform let represent the system of equations (2) in a form: ~ Xk ~ ~ 2 U k 1 k lU l 2 2 2 (4) ~ X k includes not only the force impacts on medium but also kinematic ones. Solving of algebraic equations system (4) may be writtendown in the following form: Expression (3) contains information on mass load, impacts on the medium boundary and initial conditions. If the medium motion is considered at the finite interval (0, Т) of time, the final conditions of displacements and velocities functions of the medium points at time Т are included, which represent unknown quantities; in the used method the solutions fulfill a role of “integration constants”. F j coincides with f j in area and in Fourier transform space-wise, 2 c1 In the right member of equation (4) ji s cosn, xi s U k t 0 t U j t T t T U k t 0 t U t T F c2 2 and integration: notations Let divide the left and right members of system of equations (2) by and introduce the notations: ~ ~ 2 2 2 2 X j 2 1 j k X k ~ Uk c12 2 2 2 2 2 (5) For detection of displacements it is necessary to perform an inverse Fourier transform: U j x1 , x2 , x3 , 1 4 2 ~ ~ X j 2 1 j k X k 2 2 2 2 c12 2 2 2 2 2 W e xk k dW (6) where W means a space of variables 1 , 2 , 3 , and dW d1 , d 2 , d 3 , d . For the infinite space symbols ~ X j and X j have only ~ a generalized load F j and F j . 3. Results and Discussion Let consider the method of solution of the tasks of waves propagation in the infinite elastic medium from the point-sources with application of the Fourier transform and generalized functions from the point-sources using the example of a task of the concentrated force F t effect for comparison with famous Love solution[2]. Upon effect of the concentrated force at the origin of coordinates and in х3 – direction expressions (3) take the form: X l 0 , X 2 0 , X 3 F t x3 (7) and correspondingly the Fourier transform ~ ~ X 3 F . Under such conditions the displacement components take the form: U 1 x1 , x2 , x3 , ~ 1 3 X 3 2 1 2 2 2 4 c1 W 2 2 2 2 e xk k dW , U 2 x1 , x2 , x3 , e cos r , F t 4 r c12 c1 U 0 , Ur U ~ 2 3 X 3 1 4 2 c12 W 2 2 2 2 2 2 xk k 1 4 c12 2 U r , / 2 U r r ,0 (8) Let us introduce the notation: r x12 x22 x32 . Considering radiation conditions and using asymptotic development[3] of the Fourier integrals (8) and neglecting components representing fluctuation of near-field we will get U1 2r 4 r x1x3 0 c 22 . c12 (11) For majority of rocks ratio (11) at Poisson coefficient 0,25 is approximately equal to three. The achieved method is used upon getting of functions describing wave propagation from different sources. Wave radiation patterns from such sources are provided below with the required clarifications. Propagation of stress waves from the point-sources of different types acting in the infinite elastic medium. The graphs below have been obtained with use of MATLAB program complex[5-10]. 1 3.1. Waves propagation from concentrated force cos r , F t 4 c12 r c1 u 0 , ur 1 r 1 r 2 F t 2 F t , c1 c1 c2 c2 2r U2 4 r x 2 x3 1 u 1 r 1 r 2 F t 2 F t , c1 c1 c2 c2 U3 . longitudinal motion directed along the force line ~ 2 2 2 2 2 1 32 X 3 2 2 2 2 2 W e xk k dW . sin r F t 2 4 r c2 c2 (10) An interesting fact shall be pointed out: over all distances from the point of force cross motions perpendicular to the force line / 2 exceed the dW , U 3 x1 , x 2 , x3 , In consequence of symmetry of displacements and stresses about axis х3 it is possible to get rather convenient expressions, if a spherical coordinate system going through the coordinates center is applied and shall be defined as an angle between the radial coordinate and positive axis х3 sin r F t 2 4 c2 r c2 (12) . 2r 4 r x32 1 1 r 1 r 2 F t 2 F t c1 c1 c 2 c 2 . (9) We point out that it is possible to get more complete transforms of integrals (8). In such case there are solutions fully coinciding with Love solutions[2] for the concentrated force in the infinite space. This article deals with propagation of stress waves from the point-sources of different types in the infinite space based on the asymptotic development of Fourier integrals. а – P waves FIGURE 1: Diagrams concentrated force. b – S waves of waves propagation from 3.2. Propagation of waves from combination of two forces (double force without moment) ur 2h cos2 r , F t 4 c13 r c1 u 0 , u (13) 2 h sin cos r F t . 3 4 c2 r c2 а – P waves b – S waves FIGURE 4: Diagrams of waves combination of two double forces. propagation from 3.4. Propagation of waves from combination of three couples of forces without moment directed in parallel with three orthogonal axes ur а – P waves FIGURE 2: Diagrams of combination of two forces. u 0 , b – S waves waves propagation 2h r , F t 4 c13 r c1 from (15) u 0 . а – P waves FIGURE 3: Clarification to diagram of waves propagation from combination of two forces without moment. FIGURE 5: Diagrams of waves propagation combination of three couples of forces. from 3.3. Propagation of waves from combination of two double forces (two double force without moment) 3.5. Propagation of waves from two couples of forces ur 2h sin 2 r F t , 3 4 c1 r c1 u 0 , u 2 h sin cos r F t c . 4 c23 r 2 ur 0 , (14) u 2h sin r F t 4 c23 r c2 u 0. , (16) а – S waves FIGURE 6: Diagrams of waves propagation from two couples of forces. 3.6. Propagation of waves from combination of two couples of forces with sum of moments equal to zero 2 h sin cos sin 2 r ur F t c 4 c13 r 1 , a-P waves 2 h sin cos2 sin 2 r u F t c 4 c23 r 2 2 h sin cos sin cos r u F t c 4 c23 r 2 b- S waves FIGURE 8: Diagrams of propagation of waves created by uniform pressure. 3.8. Propagation of waves created by tangential stress at alveole contour in length d ur 0 , (17) u a2d r t 4 c 2 r c1 , (19) u 0 а – P waves b – S waves FIGURE 7: Diagrams of waves propagation from combination of two couples of forces with sum of moments equal to zero. 3.7. Propagation of waves created by uniform pressure in alveoleat a sections in length d a - S waves c2 a2d r 1 2 22 cos2 p t 4 c1 r c1 c1 u 0 , FIGURE 9: Diagrams of propagation of waves created by tangential stress at alveole contour in length d. ur a 2 d sin cos r u p t . 2 c2 r c2 (18) 3.9. Propagation of waves created by tangential stress applied to alveole contour at section in length d The forms of waves propagating from different sources comply with the law of force variation and are derivatives of functions describing variation of forces and stresses. 2 a d cos r p t 2 4 c1 r c1 u 0 ur u 0 u 2 a d sin 4 c22 r (20) r t . c2 Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] а – P-waves b – S-waves FIGURE 10: Propagation of waves created by tangential stress applied to alveole contour. Remark: Derivative with respect to all expressions above between the brackets is marked with a prime. 4. Conclusions Solutions of stress waves propagation in elastic medium from different sources have been provided. The point-sources and their combinations have been selected in such a way to model the influence of machines and processes on soil body in case of shielddriven pipes (tunnels). P-waves and S-waves propagate from each source of waves. The waves propagate from concentrated force, the forms of these waves comply with the law of force variation. Wave amplitude decreases as 1 due to r radiation in space. Upon propagation of waves from double force without moment the wave forms represent a derivative of functions describing the law of force variation. The wave amplitude also decreases as space. 1 due to radiation in r Kurbatskii E.H. Method for solving problems of structural mechanics and the theory of elasticity, based on the properties of the Fourier image of finite functions: the Dissertation for the degree of Doctor of Technical Sciences. MIIT, Moscow, (1995) [2] Myshkis A.D. Mathematics for technical colleges. Special courses, 15(6), 632-644, (1996) [3] Verruijt, A., Booker, J.R. Surface settlements due to deformation of a tunnel in an elastic half plane. Geotechnique 46(4), 753-756. (1996) [4] Loganathan, N., Poulos, H.G. Analytical prediction for tunnelling- induced ground movements in clays. J. Geotech. Geoenviron. Eng. ASCE, 124(9), 846-856. (1998) [5] Lee, K.M., Rowe, R.K., Lo, K.Y. Subsidence owing to tunnelling I: estimating the gap parameter. Canada. Geotech. J. 29, 929-940. (1992) [6] Park, K.H. Analytical solutions for tunnellinginduced ground movements in clays. J. Tunnelling and Underground Space Technology, 20(5), 249-261. (2005) [7] Franzius, J. N. Behavior of Buildings due to Tunnel Induced Subsidence, Ph.D. Thesis, Department of Civil and Environmental Engineering, Imperial College of Science, Technology and Medicine, London. (2003) [8] John, O. Bickel, Thomas R. Keusel, and Elwyn H. King. Tunnel Engineering Handbook, Chapman and Hall, 544-562. (1996) [9] Leblais, Y. Recommendations on Settlements Induced by Tunnelling. Association Franзaise des Travaux en Souterrain (AFTES). 132-134. (1995) [10] Hulme, T.W., Shirlaw, J.N., Hwang, R.N. Settlements during the underground constructions of the Singapore MRT, Tenth Southeast Asian Geo- technical conference, Taipei, 16-20. (1990) [11] Wang, J.G., Kong, S.L., Leung, C.F. Twin Tunnels-Induced Ground Settlement in Soft Soils. Geotechnical Engineering in Urban Construction, Tsinghua University Press, 241244. (2003)

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