STRESS-STRAIN STATE OF PARALLEL CYLINDRICAL TUBES FILLED WITH LIQUID UNDER HARMONIC LOADS

ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 889 STRESS-STRAIN STATE OF PARALLEL CYLINDRICAL TUBES FILLED WITH LIQUID UNDER HARMONIC LOADS B.S. Raxmonov Prof. DSc, Urgench State University named after Abu Rayhan Beruni. U.I. Safarov Dots. PhD, Tashkent Institute of Architecture and Civil Engineering. https://doi.org/10.5281/zenodo.18033085 Abstract. The study examines the stress–strain state of parallel cylindrical tubes filled with liquid. The problem is solved in a bicylindrical coordinate system under the action of harmonic waves. An analytical solution is obtained in terms of special Bessel and Hankel functions, along with numerical results. A parametric analysis of the dynamic stress coefficient is carried out. Keywords: cylindrical tube, liquid, harmonic waves, bicylindrical coordinate system, special functions. Some Fundamental Relations of the Theory of Elasticity. This section presents several basic equations of the theory of elasticity in curvilinear coordinates. It is known that, according to the static theory of elasticity, the Lamé equation in vector form has the following form [1, 2, 3]: 0=fQ+urotrot-u )2(    divgrad (1) where λ and μ are the Lamé coefficients, defined by the formulas:      E ()( )1 2 1 ,   E 2 1( ) , u  - displacement vector, fQ - vector of body forces. The operators appearing in equation (1) for a right-handed system of curvilinear orthogonal coordinates are defined as follows: grad qiqiqi         1 1 1 11 1 1 22 2 33 3 3 2    , rotu qG 1 divu quq quq quq q                       1 1 1 11 2 2 22 3 3 33       G q i q i q i u q u q u q  11 122 233 3 1 2 3 111 222 333          where αᵢ - are the curvilinear coordinates (i = 1…3), qᵢⱼ — components of the metric tensor, defined by the formula: qx ij k k i    1 3    xk j xₖ — Cartesian coordinates (k = 1…3), q — the square of the Jacobian determinant of the transformation from the Cartesian coordinate system to the curvilinear coordinate system. ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 890 For orthogonal curvilinear coordinates only the diagonal components of the metric tensor qij - are nonzero. In this case, q qii i    1 3 , and the fundamental differential quadratic form is given by the formula: ds q d ii i 2 2 1 1 3    . To determine the stress state of the soil and to formulate mixed boundary conditions, it is necessary to use relations that express stress in terms of displacement. We employ the geometric equations derived by Novitsky V.            3 1 2 2 2 1 jj j j i i i i i ii h u h h h u      (2)     ij i j i i i i j i j j h h hu hhu h                  1 2 2 2 i j, j = 1,3 In addition, we use the constitutive equation (Hooke’s law) [2].   ij ij kk ij k    2 1 3 (3) Substituting (2) into (3), we obtain:                   3 1 3 1=j j j 2 2 i i 3 1=j j j 2 2 k h u 2 1 h u 2 h u 2 1u kj i iij k kkk ij h h h hh           (4a)      ij i j i i j i h h h h       2 2 u h u h i j i i j j , (4b) where iiiqh  2 . Now we formulate the linear elasticity problem for the computational models in cylindrical coordinates r, θ and z. We use the components of the displacement vector ur, uθ and uz as the unknowns. The cylindrical coordinate system is related to the Cartesian coordinate system by the following relations: x = r cos  ; y = r sin  , z = z, ds2 = dr2 + r2d  2 + dz2. (5) Using formula (5), we obtain h h q q q r 1 2 3 2 11 33 22 2 1     , h22 2 As the coordinates αi (i = 1, 3), we use:  1 = r,  2 =  ,  3 = z (6) Substituting (5) and (6) into (1), and then substituting the resulting expression into formula (4) and taking into account the above, we obtain the following system of Lame equations in cylindrical coordinates: ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 891 ()( ) ( ) ( ) ( ) ( )( ) ( ) ( ) , ( ) ( ) ( ) ( ) ( ) ( )                                                2 2 3 2 0 2 0 2 2 2 22 2 2 uru u ru u rururu uru u ruru ruru rrr rzz r z zz r r rr zz r r z z r , ( ) ( ) ( )( ) ( )( ) ( ) ( )            uru u u ruru zrr zzz rrr z r z        22 0 (7) where the indices r, θ and z outside the brackets denote partial derivatives with respect to the corresponding coordinates. The boundary conditions on the outer surface of the pipe correspond to the condition of perfect contact with the soil, while the inner surface is free: ,0 ,0 ,0: , , , , ,: 1120 ,212121 22121    rzrrr rzrzrrrrrr zzrr Rr uuuuuuRr      (8) where the indices ‘1’ and ‘2’ denote, respectively, the materials of the surrounding medium and the pipe. The boundary conditions that ensure the equality of the normal components of the velocities of the fluid and the shell are ar V t u += )n (r2     (9) where v  - the velocity of a fluid particle; n - is the normal to the surface at r = a , w - is the radial displacement of the shell. To fully close the problem formulation, it is necessary to supplement conditions (8) and (9) with the conditions at infinity.  u0 . At R x y z     2 2 2 (10) supplemented by certain radiation conditions. For non-stationary problems, the radiation conditions require the fulfillment of the causality principle, and the medium must exhibit no displacements outside the region bounded by the leading front of the waves generated by the vibration sources. ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 892 Figure 1. Computational scheme. Consider the problem of the dynamic theory of linear elasticity concerning the effect of seismic waves on pipes laid in a high embankment in two parallel lines and filled with an ideal compressible fluid. We consider the case when the incident wave strikes perpendicular to the axis connecting the centers of the pipes and also perpendicular to the longitudinal axes of those pipes. The computational model is shown in Figure 1. The bicylindrical coordinate system is related to the Cartesian coordinate system by the following relations: x = (asin  ) / (ch  -cos  ), y = (ash  ) / (ch  cos  ), z = z (11) where a is half the distance between the points η=−∞ and η=∞. Then, substituting (11) into (5, 6), and substituting the resulting expressions into (6), we obtain the following form: ds ach ach ddz 2 2 2 2 2 2 2 2        ( cos ) ( cos )      d (12) Using formula (11), we obtain h h q q a ch 1 2 2 2 11 22 2 2       ( cos ) ,  h q 3 2 33 1  (13) Assuming that α1=ξ, α2=η, α3=z, and substituting (12) and (13) into (1) - (11), and taking into account that the problem is planar, we obtain the following Helmholtz equation in bipolar coordinates:     ach v v k v     2 2 2 0( cos ) ( ) ( )    (14) where sin cos sin sin          ch e n e n n n n               2 1 > 0 2 < 0 n=1 (15) Equation (14), after some transformations, reduces to the form ( ) ( ) ( )v v kae v        2 0 2 (16) We will seek the solution of equation (14) in the form of a series:   v v n v n n a n b iwt n      ( )cos ( ) sin     0 (17) ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 893 Substituting (17) into (16) and equating the coefficients of the corresponding harmonics, we obtain the following ordinary differential equation:   v kae n v n n "( )    2 0 2 2 (18) By the standard substitution v z t n( ) ( )  , t = exp () we reduce (18) to a Bessel equation of the form t z tz k a n z 2 2 2 2 4 0" ' ( )    (19) which has a particular solution in the form of the cylindrical function Z (2ake−η), and the solution of the Helmholtz equation takes the following form:       A Z ake n e n n iwt n ( )cos2 0  ( 20)       B Z ake n e n n iwt n ( )sin2 0  Now we impose the boundary conditions. For this we use condition (20) and the substitution r = η and θ = ξ. Taking into account the obtained relations, we will seek the solution of the boundary-value problem for the case when a P-wave (compressional) and an SV-wave (shear), incident perpendicular to the y-axis, strike the two underground pipes. The wave potential is given by: ( ) .i i x iwt Ae (21) To express equation (21) in the form of equation (20), we rewrite (21) using (12) in bipolar cylindrical coordinates.  1 2( ) exp( ) sini ik a e Ae iwt   (22) By expanding the second factor of expression (22) into a Fourier series (in complex form) and performing some minor transformations, we obtain the final expression for the potential of the incident P-wave:      1 1 0 ( ) ( )cos i n n iwt n A J n e   (23) where τ = 2 a exp (±η), and for the potential of the incident SV-wave: The remaining potentials in (20), by analogy with (23), have the following form:                03 )1()( 3 02 )2( 2 )1()( 2 02 )2( 2 )1()( 2 cos)( sin)()(,cos)()( n iwt nn r n iwt nnnn r n iwt nnnn r enJG enHFHEenHDHC   (24) The dynamic stress–strain state is expressed in terms of the potentials φ₁ and ψ₂:     )()( iii u     )()( iii u , )( )( 3 1 3     iwu (25)     )(5,0sin)(5,02 2 i    iiii d 33333   iw ,     sin)(5,05,02 2 ii i e a 1 2 2, ; / . ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 894 By substituting (24) and (25) into (8), we obtain the final solutions to the problems of the incidence of Pand SV-waves on two underground pipes, respectively. The arbitrary constants An, Bn, Cn and others are determined from a system of algebraic equations with complex coefficients. [C]{q}={} where C is a determinant of order 12×12, whose elements are Bessel and Hankel functions of the first and second kind of order n; q is the column vector of unknown quantities, and ρ is the right-hand-side vector. The system of algebraic equations with complex coefficients is solved using the Gauss method with pivot selection. The dynamic stress–strain state in the case of an incident shear wave acting on two underground pipes is also written in bipolar coordinates in an asymptotic form: u w u u z z i z zi z   , ( ) , ( )          (26) As the boundary conditions, we use condition (23) and the substitution r = η. The final solution of the problem for the case of an incident SH-wave on two pipes has the following form:                                     02 )2( 2 )1( 022 01 )1( 1011 0 0 2 )2( 2 )1( 20221 )1( 11011 0 0 2 )2( 2 )1( 021 )1( 101 ;sin)()(;sin)()( ;cos)()(;cos)()( ;cos)()(;cos)()( n iwt nnnnz n iwt nnnz n n iwt nnnnrz iwt nnnnrz n n iwt nnnnz iwt nnnnz enkHCkHBnwenkHAknw enkHCkHBkwenkHAkJkw enkHCkHBwuenkHAkJwu     (27) The unknown coefficients An,Bn,Cn are determined from the boundary conditions. Let us consider the determination of the dynamic stress - strain state of a cylindrical pipe under the action of harmonic waves. To solve the stated problem, the addition theorem is used. Addition theorems for cylindrical wave functions were derived in works [4, 5, 6]. Let there be two different polar coordinate systems (rg, θg) and (rk,θk) (Fig. 3), whose polar axes have the same direction. The coordinates of the pole θk g-system are Rkq, θkq, such that the following equality holds. Z R e Z gkg i k kg    (28) Then the addition theorem takes the following form:      pkqkkk pni kqpn in qn RriprTpeRberb kqq),exp()()()( )(        pkqkkkp pni kqpn in qn RriprbeRJearb kqq),exp()()()( )(   (29) Formula (28) makes it possible to transform the solution of the wave equation (1) from one coordinate system to another. Let us consider the analysis of a long underground multi-line pipeline under seismic excitation within the framework of a plane problem of the dynamic theory of elasticity. ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 895 In doing so, we examine the case of stationary diffraction of plane waves by a series of periodically arranged cavities reinforced with rings and containing an ideal compressible fluid inside. The solution to the posed problem will be obtained using the potential method. The boundary conditions are given by (8). The form of the incident potential also remains unchanged. The potentials of the waves reflected from the pipes, after applying the addition theorem and taking into account the periodicity of the problem, will have the following form:        1 1) 1 1 0 ( ) ( ( ) ( ) ( ) , r iwt n n n n n in e A H r S J r e              1 1 1 0 ( ) (1) ( ) ( ) ( ) , r iwt n n n n n in e B H r J r e        S A E e H m e H m n p p im n p im n p mp               (1)(1) ( ) ( ) , 1 1 10 (30)   Q B E e H m e H m n p p im n p im n p mp               (1)(1) ( ) ( ) , 1 1 10 where: ξ = k δ cos γ , δ - is the distance between the centers of the pipes. The potentials of the refracted waves in the pipes are written in the form          2 1 1 2 2 0        e E C H r D H r e i m w n n n n n in n ( ) ( ) ( ) ( ) ( ) ( ) ,          2 1 2 2 0        e E E H r F H r e i m w n n n n n in n ( ) (1) ( ) ( ) ( ) ( ) , (31) and the velocity potential in the ideal compressible fluid       3 3 0      e E G J r e i m w n n n in n ( ) ( ) ( ) , (32) The unknown coefficients An-Gn are determined by substituting (29)-(32) into (8). As a result, an infinite system of linear equations is obtained, which is solved using an approximate reduction method, provided that the following relation is not satisfied k n  ( cos )1 2 The general description of the program is intended for multi-line pipes in an embankment in the case of seismic waves incident perpendicular to the axis passing through the centers of the pipes. Figure 2. Diagram for the addition theorem. ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 896 The input data include the minimum necessary parameters: the elastic properties (E and ν) of the embankment soil and the pipes; the density of the soil, the pipe, and the fluid filling it; the inner and outer radii of the pipes; the predominant period of soil particle oscillation; the coordinates of the point at which the stress–strain state is evaluated; and the seismicity coefficient. Using a special option, it is possible to perform calculations for pipes filled with an ideal compressible fluid as well as for empty ones. The cylindrical Bessel and Hankel functions are computed using known formulas. The system of linear equations is solved by the Gauss method with pivot selection. Influence of the distance between the pipes. Table 1 presents the values of the coefficient       max max ( / ( )  rr A2 2 of the maximum radial soil pressure on the pipes for different distances d between them in the case of an incident P-wave. The following parameters were assumed: the P-wave number αr =1.0; the inner and outer radii of the pipes R0=0.8 m and R=1.0 m; the predominant period of soil particle oscillation T=0.2s. The properties of the embankment soil are: Lamé constants λ1=8.9 MPa, μ1=4.34 MPa, density ρ1=1.74 kN. The properties of the pipe material are: λ2=8690 MPa\, μ2=12930 MPa and density ρ2=2.55 kN*s2 /m4. )( tkxi Ае     Figure 3. Computational scheme. Table.1 The value of the dynamic concentration coefficient for different distances between the pipes in the case of an incident P-wave D/d 0,5 1,0 2,0 4,0 max 1,68 1,76 1,61 1,60 From Table 1, it follows that at first, as the distance between the pipes increases within the range 0.5 ≤ d / D ≤ 1.0, the coefficient ηmax increases slightly-by about 5%. With a further increase of d / D > 1.0, it decreases more sharply-by about 10%. When d / D > 2.0, the value of ηmax stabilizes, i.e., it practically does not change, and for l ≤ 4.0 it is close to the ηmax value for a single pipe according to the calculations. Therefore, the mutual influence of the reinforced-concrete pipes in a multi-line arrangement is significant when the distance between them satisfies d≤4.0, ISSN: 2181-3906 2025 International scientific journal «MODERN SCIENCE АND RESEARCH» VOLUME 4 / ISSUE 12 / UIF:8.2 / MODERNSCIENCE.UZ 897 which leads to an increase in the maximum dynamic soil pressure on them compared to a single pipe. This increase in the coefficient ηmax is associated with the superposition of waves reflected from several surfaces of the multi-line pipes. The non-monotonic increase of the coefficient ηmax with a decrease in the spacing d / D is, in our opinion, related to the interference of waves superimposed after reflection. This phenomenon is extremely important in the practical design of seismic underground multi-line pipelines, as it allows selecting an optimal spacing between the pipes at which the dynamic pressure under seismic loading is minimal. For example, according to Table 1, such a spacing is d=0.5D. It is known, for comparison, that under static loading the opposite trend is observed: the soil pressure on multi-line pipes is lower than on a single pipe. In addition to the above, when analyzing the influence of the distance between pipes on their stress–strain state, one must take into account relation (28) (the so-called “sliding points”), at which a significant increase in dynamic stresses is observed near the pipe-resonance. This phenomenon, known in optics as the Wood anomaly, is a characteristic feature of multi-line pipelines and cannot occur in a single-line pipeline. From the standpoint of design practice, it is necessary to know at what spacing the pipes can be laid so that the dangerous resonance phenomenon does not occur. This question is answered by relation (27). Let us analyze this relation for the case of incident Pand SV-seismic waves acting on the underground pipeline. Table 2 presents the dependence of the maximum clear spacing between the pipe centers dmax, at which resonance does not occur, on the angle of incidence γ of the seismic waves. Table.2 Dependence of the distance Dmax on the angle of incidence γ . Degree, 0 30 45 60 70 80 90 Dmax, M 5,0 5,36 5,86 6,66 7,45 8,52 10,0 From Table 2, it follows that the smaller the angle of incidence of the seismic wave on the pipeline, the closer the pipes must be placed to each other. Thus, the occurrence of resonance in multi-line pipelines can be avoided by selecting an appropriate distance between them, thereby ensuring the seismic resistance of the pipeline. Influence of the type of seismic excitation (P-, SV-, or SH-wave). Table 3 presents the values of ηmax - the maximum radial soil pressure on the pipes - for the incidence of Pand SV-type seismic waves at various distances d between the pipes. In this case, βr = 2 was assumed. Analysis of the data in Table 3 shows that for d/D < 4,0 the values of the coefficient ηmax for Pand SV-waves are essentially in opposite phases. That is, at d/D = 1.0, the maximum seismic impact of the P-wave is 27% higher than that of the SV-wave, at d/D = 2.0 it is 7% lower, and at d/D = 4.0 it becomes higher again, but only by about 1%. As the distance between the pipes increases, the difference between these impacts decreases, and at d/D = 4.0 it practically disappears altogether. In addition, it should be noted that under SV-wave excitation, the values of ηmax for different distances between the pipes exhibit a scatter 2.5 times larger (up to 25%) compared to P-wave excitation (up to 10%). Thus, the phenomenon of “local resonance” is more strongly pronounced for seismic excitation in the form of an SV-wave.