VIBRATIONS OF THE GROUND SURFACE AT BLASTING WORKS ON THE CONSTRUCTION OF TUNNELS Текст научной статьи по специальности «Строительство и архитектура»

VIBRATIONS OF THE GROUND SURFACE AT BLASTING WORKS ON THE CONSTRUCTION OF TUNNELS

1Ablokulov Sherzodjon Zokir ugli, 2Choriev Mirjalol

1Senior teacher of the "Higher Mathematics" department of the Tashkent Institute of Chemical

Technology

2Doctoral student at the Department of Higher Mathematics, Tashkent Institute of Chemical

Technology https://doi.org/10.5281/zenodo.10041298

Abstract. This work examines vibrations of the ground surface at blasting work on the construction of tunnels. To estimate the levels of vibration of the ground surface under explosive action inside a cylindrical cavity, to a first approximation we consider the plane problem of the theory of elasticity. As a result of this impact, longitudinal and transverse waves will propagate in the soil. To solve the problem, the reciprocity theorem is applied, for dynamic problems of the theory of elasticity, and the results obtained are compared by wave dynamics methods. The main goal of the work is to determine the stressed deformed state of pipelines under the influence of seismic waves. The resulting solution allows us to determine the levels of vibrations of the soil surface that occur during explosive effects inside the cylindrical cavity of the elastic half-space. The results can be used to assess the levels of ground surface vibrations that occur during the construction of shallow tunnels using the drill-and-blast method.

Keywords: Mechanical system, waves, vibration, cylindrical shells, solid

Introduction. To perform a computational analysis of the dynamic state of an underground pipeline, it is considered as cylindrical bodies located in an elastic medium [1,2]. The blast load can be described by a difference in form [3]. When solving the problem, the plane problem of elasticity theory is mainly used [4,5]. The calculation uses the finite element method in the form of displacements [6,7]. The displacements of the nodal points of the finite element model are taken as the main unknowns. At the same time, to increase the efficiency of calculations, methods were used that take into account the features of matrix operations (symmetry of matrices, the presence of a large number of zero elements, etc.). The entire mass of the pipeline is considered concentrated in the nodes of the calculation model, while the distributed mass of the segment is divided equally between its beginning and end [8,9]. Calculations for seismic impact are performed in many works using the dynamic analysis method or the linear-spectral method [10,11]. In the case of calculations for one group of spectra, the displacements of all fixed nodes of the calculation model are considered the same (the "rigid" platform hypothesis). In this case, the seismic load can be specified for three mutually perpendicular directions in the form of accelerograms or response spectra [12,13]. This work, unlike others, uses numerical and analytical methods to comparatively evaluate the effectiveness.

Statement of the problem and solution methods. In underground construction and, in particular, in tunneling, excavation in hard rock is most often carried out using the explosive method [1]. Explosive effects are characterized by a large release of energy in a short period of time. For example, when one kilogram of TNT explodes with a charge length of 0.25 meters, over J is released in 40 microseconds. Stress waves propagating in the ground from the explosion site can reach the surface, which can lead to damage or destruction of ground objects. To estimate the levels of vibration of the ground surface under explosive action inside a cylindrical cavity, to a first approximation we consider the plane problem of

the theory of elasticity. We will assume that as a result of an explosion inside a cylindrical cavity, uniform pressure acts on some sufficiently long part of it

Fig. 1. Loading diagram to determine stress deformed state

As a result of this impact, longitudinal and transverse waves will propagate in the soil. To determine vibrations of the ground surface, we apply the reciprocity theorem and use well-known analytical solutions to the problem of the propagation of waves in an elastic half-space under the influence of concentrated waves applied to the surface. Let us consider two states of an elastic halfspace with a cylindrical cavity. First state (I): a concentrated force is applied to the surface of the elastic half-space F (t) (Fig. 1). In order to avoid singularities, we will assume that the force is distributed over a small area. Second state (II): the stress-strain state of the elastic half-space is created by uniform pressure applied to the inner surface of the p (t) cylindrical cavity. To determine the displacement of ground points during the propagation of longitudinal and transverse waves from a vertical harmonic force applied to the surface of an elastic half-space F0e'at, we use the solution of G. Miller and N. Persey [2]:

u =

F cosp[1 - 2S2 sin2 p]e

-irnrla irnt

2npa2r {[1 - 2S2 sin2 p]2 + 4S3 sin2 pcosp[1 - S2 sin2 p]112}

(1)

u¡ = 0

(2)

-F sin < cos - sin2 p]112 e~imrlßeimt

.2 „n2

uv =

P --o2

npß2r {[1 - 2 sin2 <]2 + 4 sin2 < cos <[S2 - sin2 <]112}

(3)

where a is the speed of longitudinal waves in the soil; fi - speed of transverse waves into

P

the ground; S = — - speed ratio, a

r = h + x , cos < = h l r,sin <p = x l r.

To determine the displacement of ground points during the propagation of longitudinal and transverse waves from a horizontal harmonic force applied to the surface of an elastic half-space

Geiat, we use the solution of J. Cheney:

h _ GS cos 0 sin P cos P[1 -Ô2 sin2 Pf2 e~larlaelmt U " npa2r {[1 - 2S2 sin2 p]2 + 4S3 sin2 p cos p[1 - S2 sin2 p]1/2} (4)

h -G sinOe-lar/ßelat

U0 = ° - „2--(5)

2npß r

h

G cos 0 cos p[1 - 2 sin2 p]e

- iwr/ß iwt

U(p IxpP2r {[1 - 2 sin2 p]2 + 4 sin2 p cos (p[82 - sin2 p]1/2} (6)

Vertical vibrations of the ground surface transmitted by longitudinal waves during an explosion in a cylindrical cavity. The average radial displacements of an unsupported cylindrical cavity from longitudinal waves generated by a vertical force are determined by the formula

Auva = Aur + Aup + Aue (7)

which, after substituting expressions for average radial displacements:

b " A a v2 b .

Aur - —pa\ and Aue ----pa\

E ^ 1 -vK 1 -v E

takes the form:

h y y2 h

Aua = Aur + Au + Aue = pa\ ---) or Auar = ap\ (8)

£1-1/1-1/ 2 ju

where p is the soil shear modulus (determined by the formula: jU = 01 p )

Using expression (1) and performing the necessary transformations, we reduce the resulting equation to a more convenient form [4]:

- ior/a _iwt

v__(i&)bF0 cosp[1 - 2Sz sin2 p]e~i"r/ae_

Ua " 4njar {[1 - 282 sin2 p]2 + 4S3 sin2 p cos p[1 - 82 sin2 p]1/2} (9)

Let us denote: P0 - the amplitude value of the pressure during an explosion in a cylindrical

cavity for a harmonic component with frequency co, Aav - the vertical movement of the ground

surface, d - the distance over which the explosion pressure extends along the cylindrical cavity, r - the distance from the point at which vibrations are determined to the axis of the cylindrical cavity.

In accordance with the reciprocity theorem we have:

2nbdP, Aua= F0 AaV (10)

What follows:

_ P0b2 (ig)d cos p[1 - IS2 sin2 p]e"aem

av~ 2juar {[1 - 2S2 sin2 p]2 + 4S3 sin2 p cosp[1 - S2 sin2 p]1/2}(H)

Horizontal vibrations of the ground surface transmitted by longitudinal waves during an explosion in a cylindrical cavity.

The parameters of horizontal vibrations during an explosion in a cylindrical cavity are determined in the same way as vertical ones, except that instead of function (1), it is necessary to use function (4). Omitting intermediate calculations, we present the final result

_ pb2 (ia)dö cos 0 sin p cos p[1 -S2 sin2 p]1/2 e'aem ah " juar {[1 - 2S2 sin2 p]2 + 4S3 sin2 p cos p[1 - S2 sin2 p]1/2}

Taking into account that horizontal and vertical vibrations of the soil surface, transmitted by longitudinal waves, which arose during an explosion in a cylindrical excavation, occur in one phase, the amplitude of the vibrations can be determined by the formula:

Aa=K+Alh (13)

Vertical vibrations of the ground surface transmitted transverse waves during an explosion in a cylindrical cavity. The average radial displacements of an unsupported cylindrical cavity from transverse waves generated by a vertical force are determined by the formula:

AuV = Aur + Au^ + Au^ (14)

which, after substituting expressions for average radial displacements:

b - A v2 b

Au=-pf3i >A u0=—--Pfi,

* E 1 -v E 1 -v E

takes the form:

Auvfi = Aur + Au + Au0 = t~ ) (15)

1 - V 1-1/

Using expression (7.3) and performing the necessary transformations, we reduce the resulting equation to a more convenient form:

A v _ _ Q)bF0 sinpcosp[S2 - sin2 p]1/2 e'^/peiat

Up " 2^/upr {[1 - 2 sin2 p]2 + 4 sin2 p cos p[S2 - sin2 p]1/2} (16)

Let us denote: A^ the vertical movement of the soil surface during the propagation of

transverse waves, all other designations: p, d and r leave them unchanged. In accordance with the reciprocity theorem we have:

2nbdP0 Auvp= F Av (17)

What follows

pb2 (im)dS sin p cos p[S2 - sin2 p]1/2 e'iwr/ßei(0t jßr {[1 - 2 sin2 p]2 + 4 sin2 p cos p[S2 - sin2 p]1/2}

Horizontal vibrations of the ground surface transmitted transverse waves during an explosion in a cylindrical cavity.

Distance x(m)

Fig. 2. Levels of vertical displacements of the ground surface transmitted by longitudinal waves during an explosion in a cylindrical cavity

Distance x(m)

Fig. 3. Levels of horizontal displacements of the ground surface transmitted by longitudinal waves during an explosion in a cylindrical cavity

Horizontal vibrations transmitted by transverse waves during an explosion in a cylindrical cavity are determined in the same way as vertical ones, but here we use function (6). Omitting intermediate calculations, we present the final result:

_ pb (io) cos 0 cos p[1 - 2 sin2 p]1/2 e ~lor/Peimt

ph ~ 2vPr {[1 - 2 sin2 p]2 + 4 sin2 p cos p[S2 - sin2 p]1/2} (19)

Considering that horizontal and vertical vibrations of the ground surface transmitted by transverse waves that occurred during an explosion in a cylindrical mine occur in one phase, the amplitude of the oscillations can be determined by the formula:

Ap =^IAX+A% (20)

An example of evaluating the vibrations of the ground surface during an explosion in a cylindrical cavity.

As an example, we will determine the movement of the soil surface from an explosion in a cylindrical cavity at the following values of the initial data [5]: а = 600м / сек - the velocity of longitudinal waves in the ground, p = 350м / сек - the velocity of transverse waves in the

ground, р = 1700кг / M - the density of the soil, у = 0.0 - the coefficient of internal friction of the soil material, h = 20 м - the depth of the cylindrical cavity, b = 2.8 м - the radius of the cylindrical cavity, X - the distance from the point at which the vibration levels are determined before the projection of the tunnel axis onto the ground surface, P0 = 2МПа the amplitude value of the harmonic component of the force with frequency C = 200 Hz, d = 10м - the section

of the cylindrical cavity on which the pressure acts. The obtained solutions for harmonic forces can be used for arbitrary effects using the Fourier transform. Conclusion.

Based on the results obtained, it was found that an increase in the thickness of the filler has a particularly significant effect on the change in the complex phase velocity at relatively small thicknesses of the filler. With an increase in the order of the mode of torsional oscillations, the density of the arrangement of curves on the plane of phase velocity and wave numbers increases. In the case of short waves, there is a limit value of the complex phase velocity.

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