Analysis and calculation method for water hammer wave velocity of composite pipeline

Abstract

The invention relates to a method for analyzing and calculating the water hammer wave velocity of a composite pipeline. The method comprises the following steps: determining a function relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipeline; establishing a pipeline water hammer wave velocity basic equation; accurately analyzing the water hammer wave velocity of the composite pipeline; analyzing the water hammer wave velocity of the thin-wall composite pipeline; and confirming the water hammer wave velocity calculation and application range. The method comprises the following steps: firstly, establishing a function relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipeline, and exporting a general formula for accurately calculating the water hammer wave velocity by considering various factors; on the basis, all factors influencing the water hammer wave velocity of the thin-wall composite pipe are analyzed, water hammer wave velocity analysis formulas including the water hammer wave velocity formulas of the thin-wall lining tunnel and the underground buried pipe are derived, the water hammer wave velocity analysis formulas are compared with a traditional composite pipe water hammer wave velocity formula for analysis, and use conditions are determined. According to the method, the elastic coefficients and the water hammer wave velocities of different composite pipelines are determined, so that the calculation of the water hammer wave velocities is more accurate, and a more scientific calculation method is provided for engineering design.

Description

Technical Field

[0001] The invention relates to an analytical calculation method for water hammer wave velocity of a composite pipeline, which is a hydraulic calculation method and a calculation method for water hammer wave velocity, an important parameter of a water pipe, in a water pipeline project. Background Art

[0002] The water hammer velocity of a pipeline is a key parameter for analyzing water hammer or hydraulic transition process. As early as 160 years ago, Menabrea (1858) proposed the water hammer velocity formula under the conditions of rigid pipeline and elastic water. 40 years later, Joukowski (1898) considered the elasticity of the pipeline and proposed the water hammer velocity formula of thin-walled pipeline which is commonly used so far:

[0003]

[0004] Where: a is the water hammer wave velocity in the pipeline, m / s; K is the bulk elastic modulus of the water body, Pa; ρ is the liquid density, kg / m 3 ; D is the pipe diameter (inner diameter), m; e is the pipe wall thickness, m; E is the Young's elastic modulus of the pipe, Pa.

[0005] In the 1970s, Wylie and Streeter (1978) considered the influence of pipeline support and revised the Joukowski water hammer velocity formula. At the same time, based on the research of others, they gave the water hammer velocity formula for circular tunnels, lined circular tunnels and reinforced concrete pipes. At the same time, Coleck proposed the water hammer velocity formula for concrete lined tunnels.

[0006]

[0007] Where: K c is the elastic coefficient of the inner wall of the pipe in contact with water, that is, the deformation coefficient of the inner wall of the pipe against the change of water pressure; R1, R2, R j , R j+1 are the radii of the 1st, 2nd, jth, and j+1th layers, m; E1, E2, E j 、E n are Young's modulus of elasticity of the pipes in the 1st, 2nd, jth and nth layers, Pa; e1 is the wall thickness of the first layer, m; A2 is the equivalent area of ​​the steel bars in the axial length L = 1m in the second layer, m 2 ;μ j , μ nare the Poisson's ratios of the jth and nth layers of material, respectively; the subscript j = 1, 2, 3, ..., n is the concrete lining tunnel layer number; n is the number of tunnel layers, the first layer is the innermost layer, and the nth layer is the tunnel surrounding rock (outermost layer). On the right side of formula (3): the first item is the elastic coefficient of the thin-walled steel cylinder lining layer; the second item is the equivalent elastic coefficient of the concrete reinforcement; the third item is the elastic coefficient of the concrete lining layer and the cement grouting layer; and the fourth item is the elastic coefficient of the tunnel surrounding rock layer. The shortcoming of the Coleck formula is that it does not consider the influence of the axial and radial stresses of the tunnel on the elastic coefficient components of the tunnel lining steel cylinder and reinforcement.

[0008] Lv Haiyan (2005) and Guo Qiang et al. (2022) studied the influence of fluid-solid coupling, including axial stress, on the water hammer wave velocity of thick-walled pipes. The results showed that the obtained pipe water hammer wave velocity value was smaller than that of the classic thin-walled pipe water hammer wave velocity value.

[0009] At present, with the increasing requirements for pipeline corrosion protection and water quality, many composite pipes consisting of two or more single-layer pipes tightly fitted together have been produced, such as prestressed concrete cylinder pipe (PCCP), steel-plastic composite pipe, reinforced concrete pipe, lined tunnel, lined ductile iron pipe, etc. Zhou Wenlong et al. (2006) used the following water hammer wave velocity formula for composite pipes with inner and outer lining protection:

[0010]

[0011] Where: E p is the equivalent elastic modulus of the composite pipe, Pa; D1, D2, D3 are the pipe diameters (inner diameters) of the 1st, 2nd and 3rd layers respectively; μ1, μ2 are the Poisson's ratios of the 1st and 2nd layer materials respectively; e3 is the wall thickness of the 3rd layer, m.

[0012] Zhang Ao et al. (2022) summarized the commonly used water hammer wave velocity formulas and used the following formula for underground lined composite pipes:

[0013]

[0014] Where: E s is the Young's elastic modulus of the soil; c1 is the correction factor considering the pipeline support condition; e2 is the pipe wall thickness of the second layer, m.

[0015] Comparing the Kleck water hammer velocity formula (2) (3) with the water hammer velocity of the protective layer pipeline (4) (5), the two have certain similarities. Formula (5) E p This is equivalent to taking formula (3) K c The first and third terms in (4) are different from those in (2), but D1e1 is added in (4) compared with (2). On the other hand, when (3) K c Item 1 2R j =D (pipe water section diameter) and Kc The fourth item μ n =2.5, then we can get formula (6).

[0016] The existing problem is how to systematically consider the main factors affecting the water hammer velocity of composite pipes to determine the theoretical accurate value of the water hammer velocity and the applicable scope of the traditional composite pipe water hammer velocity formula. Summary of the invention

[0017] In order to overcome the problems of the prior art, the present invention proposes a method for calculating the water hammer wave velocity of a composite pipeline. The method proposed herein, based on the experience of predecessors, proposes several methods for calculating the water hammer wave velocity of a composite pipeline, clarifies its scope of use, and corrects the mistakes and some misunderstandings of predecessors, making the calculation of the water hammer wave velocity of the composite pipeline more accurate, and providing a more precise method for engineering calculations.

[0018] The object of the present invention is achieved by: a method for analyzing and calculating the water hammer wave velocity of a composite pipeline, the steps of the method are as follows:

[0019] Step 1, determining the functional relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipeline: according to the original definition of the water hammer wave velocity, determining the functional relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipeline;

[0020] The original definition of water hammer wave velocity a in water pipeline is:

[0021]

[0022] Where: K is the bulk elastic modulus of water; ρ is the liquid density; A is the rate of change of water pressure on the inner wall of the pipe; i is the water-passing cross-sectional area of ​​the inner wall of the pipe; is the change rate of the water-passing cross-sectional area of ​​the inner wall of the pipe;

[0023] Relative expansion rate of the water-passing cross-sectional area of ​​the pipeline:

[0024]

[0025] Where: is the circumferential strain rate of the inner wall of the pipe;

[0026] When defining:

[0027]

[0028] Where: K c is the elastic coefficient of the inner wall of the pipe in contact with water;

[0029] Then K c The pipe water hammer wave velocity a0 is defined as:

[0030]

[0031] Step 2: Establishment of the basic equation for pipeline water hammer wave velocity: Establish the basic equation for the circumferential strain rate of a thick-walled cylinder and the circumferential, radial and axial stress change rates, and obtain the circumferential strain rate of the inner wall of the pipeline according to the pipeline support conditions. and the circumferential strain rate of the outer wall The basic equation is:

[0032]

[0033]

[0034] Where: E is Young's elastic modulus; c1 is a constant related to the pipe support; R i is the radius of the inner wall of the pipe; R o is the radius of the pipe outer wall; p i is the inner wall pressure of the tube; p o is the outer wall pressure of the pipe; μ is the Poisson's ratio of the pipe material;

[0035] For single-layer thick-walled pipes, Water hammer wave speed a1 formula:

[0036]

[0037] Step 3, accurate analysis of water hammer wave velocity of composite pipe: establish the basic equation of the relationship between the circumferential strain rate of the composite pipe wall in contact with water and the strain rates of other layers, interface pressure, pipe mechanical parameters and structural dimensions, and derive the general formula for accurately solving the water hammer wave velocity of composite pipe by analysis;

[0038] The composite pipe is a pipe composed of two or more single-layer pipes tightly fitted together. The water hammer wave velocity a depends on the circumferential strain rate of the pipe wall in contact with water, that is, the circumferential strain rate of the inner wall of the first layer of the composite pipe. and It is not only related to the strain rate of the first layer's outer wall, but also to the strain rates of the inner and outer walls of other layers;

[0039] Circumferential strain rate of the inner wall of layer j

[0040]

[0041] Where: c j is a constant related to the support of the j-th layer of pipeline, j is the layer number of the composite pipeline, j = n, n-1, n-2, ..., 1; n is the number of layers of the composite pipeline, which is a natural number; E j is the Young's elastic modulus of the j-th layer of pipe; E j 、E j+1is the radius of the j-th layer pipeline and the j+1-th layer pipeline; μ j is the Poisson’s ratio of the j-th layer pipeline material;

[0042] Circumferential strain rate of the outer wall of the jth layer

[0043]

[0044] At the interface between the j-1 layer outer wall and the j-th layer inner wall of the composite pipe, under the influence of water pressure, the j-1 layer outer wall will push the j-th layer inner wall outward, and the j-1 layer outer wall will produce a circumferential strain rate At the same time, the inner wall of the jth layer will push the outer wall of the j-1th layer inward, generating a circumferential strain rate As a result, the two layers are squeezed tightly together. At this time, the strain rate boundary condition at the interface between the two layers is:

[0045]

[0046] Where:

[0047]

[0048]

[0049] Where:

[0050]

[0051]

[0052] In the imperative form:

[0053]

[0054] During the pipe water hammer process, At this time, take:

[0055] w n+1 =0

[0056]

[0057] Where: w j 、w j+1 are the transition symbols of the jth layer and the j+1th layer respectively; b j,1 、b j,2 、b j,3 、b j,4 are coefficients related to the mechanical properties and structural parameters of each layer of the composite pipe material;

[0058] Elastic coefficient K of the inner wall of the composite pipe in contact with water c0 :

[0059]

[0060] in: b is the circumferential strain rate of the first layer of pipe wall in contact with water; 1,3 、b 1,4 , w2 is the coefficient and transition symbol related to the mechanical properties and structural parameters of each layer of the composite pipe material;

[0061] In summary, the calculation procedure for the water hammer wave velocity of the composite pipe is: algebraic recursion calculates w n 、w n-1 ,…,w2, and then calculate the elastic coefficient K of the composite tube c0 , and finally by the formula:

[0062]

[0063] Calculate the water hammer wave velocity; in the derivation process, the influence of the composite pipe on the circumferential strain rate is fully considered The various factors of the water hammer wave velocity calculated in this way are accurate solutions;

[0064] Step 4, analysis of water hammer wave velocity of thin-walled composite pipes: Analyze various factors of the circumferential strain rate of the thin-walled composite pipe in contact with water, and propose an analytical formula for water hammer wave velocity, including analytical formulas for water hammer wave velocity of thin-walled lined tunnels and underground buried pipes;

[0065] General elastic coefficient K of thin-wall composite pipe c1 :

[0066]

[0067] Water hammer wave velocity a1 of thin-wall composite pipe:

[0068]

[0069] Elastic coefficient K of thin-walled lining tunnel c2 :

[0070]

[0071] Water hammer wave velocity a2 in thin-walled lining tunnel:

[0072]

[0073] The elastic coefficient and water hammer wave velocity of the inner wall of underground pipeline are:

[0074]

[0075]

[0076] Where: En is the elastic modulus of backfill soil; c s is the empirical coefficient;

[0077] Step 5, confirm the water hammer wave velocity calculation and applicable scope:

[0078] Method 1: For composite pipes with thick wall layers, including lining, the general water hammer velocity formula for composite pipes is used:

[0079] For composite pipes with thick wall layers, including lining, the water hammer velocity is determined using the following water hammer velocity formula:

[0080]

[0081] Elastic coefficient K of thick-walled composite pipe c :

[0082]

[0083]

[0084] Method 2: For thin-wall composite pipes, the water hammer velocity formula for thin-walled lined tunnels is used:

[0085] Elastic coefficient K of thin-walled lining tunnel c2 :

[0086]

[0087] Water hammer wave velocity a2 in thin-walled lining tunnel:

[0088]

[0089] The advantages and beneficial effects of the present invention are as follows: First, based on the original definition of water hammer wave velocity, the present invention establishes the functional relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipeline, as well as the functional relationship between the circumferential strain rate and the circumferential, radial and axial stress change rates. Then, various factors such as the strain rate of each layer of the composite pipe, the interface pressure, the mechanical parameters of the pipe material and the structural dimensions are considered to derive a general formula for accurately calculating the water hammer wave velocity. On this basis, the various factors affecting the water hammer wave velocity of thin-walled composite pipes are analyzed, and an analytical formula for the water hammer wave velocity is derived, including the water hammer wave velocity formula for thin-walled lined tunnels and underground buried pipes, and a comparative analysis is performed with the traditional composite pipe water hammer wave velocity formula. The present invention clarifies the elastic coefficient and water hammer wave velocity of different types of composite pipes, thereby making the calculation of the water hammer wave velocity more accurate, and providing a more scientific calculation method for engineering design. BRIEF DESCRIPTION OF THE DRAWINGS

[0090] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0091] Figure 1 is a flow chart of the method described in an embodiment of the present invention;

[0092] Figure 2 is a schematic diagram of characteristic parameters of a thick-walled pipe cross section in the method according to an embodiment of the present invention;

[0093] Figure 3 It is a schematic cross-sectional view of the composite pipeline according to an embodiment of the present invention. DETAILED DESCRIPTION

[0094] Example:

[0095] This embodiment is a method for analyzing and calculating the water hammer wave velocity of a composite pipeline. The method is based on the original definition of the water hammer wave velocity of the pipeline. First, the relative expansion rate of the water hammer wave velocity and the water-passing cross-sectional area of ​​the pipeline and the circumferential strain rate of the pipe wall are established. Then, based on the basic theory of stress and strain of thick-walled cylinders, the following functional relationship was established: The basic equation of the functional relationship with the tube structure and mechanical parameters is obtained, and the calculation formula for the water hammer wave velocity of a single-layer thick-walled pipe is derived. On this basis, a set of linear algebraic equations that relate the composite pipe layer and the interlayer circumferential strain rate is established, and a general formula for accurately calculating the water hammer wave velocity of pipelines and tunnels is obtained analytically. Then, for thin-walled composite pipes, including tunnels and underground buried pipes, a simplified formula for the water hammer wave velocity of thin-walled composite pipes is obtained analytically and compared with the traditional water hammer wave velocity formula. Finally, several water hammer wave velocity calculation methods suitable for different composite pipes are given, and the specific process is as follows: Figure 1 shown.

[0096] Step 1, determine the functional relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipe: According to the original definition of the water hammer wave velocity, determine the functional relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipe.

[0097] Basic method of pipeline water hammer velocity:

[0098] The original definition of water hammer wave velocity in water pipeline is:

[0099]

[0100] Where: p i is the pipe water pressure on the inner wall of the pipe, Pa; superscript “·” = ddt; is the water-passing cross-sectional area of ​​the inner wall of the pipe, m 2 ; R i is the radius of the inner wall of the pipe; the subscript i represents the inner wall of the pipe.

[0101] like Figure 1 As shown, due to the elasticity of the pipe wall, the water pressure p on the inner wall of the pipe is i and external pressure p oUnder the action of i , the change in the cross-sectional area of ​​the pipe is △A i :

[0102]

[0103] The second term on the right side of the above equation is a higher-order trace compared to the first term, that is, so:

[0104] △A i ≈2πR i u i (8)

[0105] When water hammer occurs, p i and p o Changes over time, causing △A i Changes with time. Derivative of formula (8) yields:

[0106]

[0107] The circumferential strain rate of the inner wall of the pipe The relationship is:

[0108]

[0109] According to formula (9) and (10), the relative expansion rate of the pipe water flow cross-sectional area can be obtained:

[0110]

[0111] When defining:

[0112]

[0113] Substituting equation (11) into equation (7), we can get the pipeline water hammer wave velocity:

[0114]

[0115] Obviously, the greater the deformation of the inner wall of the pipe, the greater the K c The smaller it is, the smaller the water hammer wave speed is.

[0116] Step 2, establish the basic equation of pipeline water hammer wave velocity: establish the basic equation of the circumferential strain rate of thick-walled cylinder and the circumferential, radial and axial stress change rate:

[0117] Assuming that the tube is isotropic and the cross section perpendicular to the tube axis remains flat after deformation, the following basic equations can be obtained based on the strain and stress analysis theory of thick-walled cylinders:

[0118]

[0119] Where: is the circumferential strain rate; is the axial strain rate; is the rate of change of pipeline circumferential stress; is the rate of change of pipeline axial stress; is the rate of change of radial stress of the pipeline; μ is Poisson's ratio; E is Young's elastic modulus;

[0120]

[0121] Where: σ is the pipeline stress, Pa; c1 is the coefficient related to the pipeline support; r is the radius of any circle on the pipeline wall, m; p o is the pressure outside the pipe; R o is the radius of the outside of the pipe; the subscripts θ, r, and z represent the circumferential, radial, and axial directions of the pipe, respectively; the subscript o represents the outside of the pipe wall.

[0122] From formula (13), we can see that the circumferential strain rate It is not only related to the rate of change of circumferential stress And the radial stress change rate and the axial stress change rate The axial strain rate of the pipe and stress change rate Regarding the support of the pipeline, it can be divided into the following three situations:

[0123] (a) The pipe is fixed only at the upstream end:

[0124]

[0125] (b) The whole tube is fixed and there is no axial movement. According to formula (14), we can get:

[0126]

[0127] (c) All pipes are connected by expansion joints:

[0128]

[0129] The boundary condition for the rate of change of stress on the inner wall of the pipe is:

[0130]

[0131] The boundary condition for the stress change rate of the outer wall of the pipe is:

[0132]

[0133] For the pipeline support method (a), substitute equations (17), (20), (21) into equation (13) to obtain the pipeline inner wall strain rate:

[0134]

[0135] For the pipe support method (b), substitute equations (18), (20), (21) into equation (13) to obtain the pipe inner wall strain rate:

[0136]

[0137] For the pipe support method (c), substitute equations (19), (20), (21) into equation (13) to obtain the pipe inner wall strain rate:

[0138]

[0139] Observe equations (24)-(26), they can be described in the following unified form:

[0140]

[0141] Where: c1 is the support condition for the pipeline:

[0142] (a) c1 = 1 - μ2

[0143] (b) c1 = 1 - μ 2

[0144] (c) c1=1 (28)

[0145] For the pipe support method (a), substitute equations (17)(22)(23) into equation (13) to obtain the pipe outer wall strain rate; for the pipe support method (b), substitute equations (18)(22)(23) into equation (13) to obtain the pipe outer wall strain rate; for the pipe support method (c), substitute equations (19)(22)(23) into equation (13) to obtain the pipe outer wall strain rate. Similarly, the unified description form of the pipe outer wall strain rate can be obtained:

[0146]

[0147] In general, when water hammer occurs, the pressure on the outer wall of the pipe changes slightly with time and can be ignored, that is, At this time, according to equations (27)(11)(12), the water hammer wave velocity of thick-walled pipe can be obtained:

[0148]

[0149] Since all factors affecting the circumferential strain rate of the inner wall of the pipe are comprehensively considered in the derivation process and there is no simplification in the analysis process, the result obtained by equation (30) is the accurate solution of the water hammer wave velocity.

[0150] Because the inner diameter of the pipe D = 2R i ,and So formula (30) can be rewritten as:

[0151]

[0152] Formula (31) and the water hammer velocity formula for thick-walled pipes recommended by Wylie and Streeter [1] Exactly the same. Observing formula (31), it seems that the water hammer wave velocity a has nothing to do with the water pressure p, but in fact it is related, because the greater the water pressure on the pipeline, the greater the required pipe wall thickness e, and the greater the water hammer wave velocity.

[0153] For thin-walled pipes, eD<125, then equation (31) can be simplified to:

[0154]

[0155] The above formula is the commonly used formula for water hammer wave velocity in thin-walled pipes that takes into account the pipe support conditions.

[0156] For a circular tunnel, assuming that the surrounding rock is buried deep, R o >>R i When R o →∞, then from equation (31), the water hammer wave velocity in the tunnel can be obtained as

[0157]

[0158] Step 3, accurate analysis of the water hammer wave velocity of the composite pipe: establish the basic equation for the relationship between the circumferential strain rate of the composite pipe wall in contact with water and the strain rates of other layers, interface pressure, pipe mechanical parameters and structural dimensions, and analytically derive a general formula for accurately solving the water hammer wave velocity of the composite pipe.

[0159] Composite pipes are pipes composed of two or more single-layer pipes tightly fitted together, such as prestressed concrete cylinder pipe (PCCP), steel-plastic composite pipe, reinforced concrete pipe, lined tunnel, lined ductile iron pipe, etc.

[0160] like Figure 2 As shown in the figure, for the composite pipe, the water hammer wave velocity a depends on the circumferential strain rate of the pipe wall in contact with water, that is, the circumferential strain rate of the inner wall of the first layer of the composite pipe and It is not only related to the strain rate of the first outer wall, but also to the strain rates of the inner and outer walls of the other layers. In other words, it is necessary to establish a functional relationship between the strain rates of the composite pipe layers and then solve them simultaneously.

[0161] The circumferential strain rate of the inner wall of the jth layer can be obtained from formula (27):

[0162]

[0163] Where: j = n, n-1, n-2, ..., 1; n is the number of layers of the composite pipe; for the pipe support:

[0164] (a)c j =1-μ j 2

[0165] (b)

[0166] (c)c j =1 (35)

[0167] From (29), the circumferential strain rate of the outer wall of the jth layer can be obtained:

[0168]

[0169] At the interface between the j-1 layer outer wall and the j-th layer inner wall of the composite pipe, under the influence of water pressure, the j-1 layer outer wall will push the j-th layer inner wall outward, generating a circumferential strain rate At the same time, the inner wall of the jth layer will push the outer wall of the j-1th layer inward, generating a circumferential strain rate As a result, the two layers are squeezed tightly together. At this time, the strain rate boundary condition at the interface between the two layers is:

[0170]

[0171] Formula (34)(36)(37) can be rewritten as the following linear equations:

[0172]

[0173] Where: b and w are transition symbols and have no physical meaning. It can be seen from the following formula that b has a certain relationship with some parameters of the pipeline.

[0174]

[0175]

[0176] If the outermost layer of the composite pipe j = n is the tunnel surrounding rock, then R n+1 >>R n , then from formula (41) we can get:

[0177] b n,3 =(1+μ n )E n , b n,4 =2c j E n (42)

[0178] When the command formula (40):

[0179]

[0180] Where: w j+1 is the coefficient to be determined.

[0181] During the water hammer process in the pipeline, it is advisable to At this time, take:

[0182] w n+1 =0 (44)

[0183] Substituting formula (43) into formula (40), we can obtain:

[0184]

[0185] Substituting formula (43) into formula (39), we can obtain:

[0186]

[0187] For the first layer of the inner wall of the composite pipe, the circumferential strain rate is:

[0188]

[0189] Substituting formula (47) into formula (11), the elastic coefficient of the inner wall of the composite pipe in contact with water is obtained:

[0190]

[0191] In summary, the calculation procedure of the water hammer wave velocity of the composite pipe is: w2 is calculated by algebraic recursion of formula (45), and then the elastic coefficient K of the composite pipe is calculated by formula (48): c Finally, the water hammer wave velocity is calculated by equation (12). In the process of deriving equations (45)(48)(12), the influence of the composite pipe on the circumferential strain rate is fully considered. Therefore, the water hammer wave velocity calculated in this way is an accurate solution.

[0192] It should be noted that formulas (45), (48), and (12) are not only applicable to thick-walled pipes and composite pipes, but also to thin-walled pipes, tunnels, lined tunnels, etc., so these formulas are universal water hammer wave velocity formulas for pipes and tunnels. In general, the parameters involved in the formulas can be obtained in the design.

[0193] Step 4, analysis of water hammer wave velocity of thin-walled composite pipes: Analyze various factors of the circumferential strain rate of the thin-walled composite pipe in contact with water, and propose an analytical formula for the water hammer wave velocity, including analytical formulas for the water hammer wave velocity of thin-walled lined tunnels and underground buried pipes.

[0194] When the pipe diameter D j =2R j With wall thickness e j =R j+1 -R j The ratio D j e j =2R j e j ≥25, it can be regarded as a thin-walled composite pipe. The analytical solution of the water hammer wave velocity of thin-walled composite pipe is studied below.

[0195] For thin-walled tube layers:

[0196]

[0197] Since the Poisson's ratio μ of the pipe is less than 0.5, for example, concrete μ = 0.1-0.15, steel pipe μ = 0.25, cast iron μ = 0.25, plastic μ = 0.33, polyethylene μ = 0.46, epoxy resin μ = 0.38, then substituting equation (49) into equation (41) yields:

[0198]

[0199]

[0200]

[0201]

[0202] From formula (50), we can see that for thin-walled tube layers:

[0203] b j,1 =b j,2 , b j,3 =b j,4 , b j,2 =b j-1,3 (51)

[0204] For j=n layers, w n+1 =0, according to formula (44)(46),

[0205]

[0206] For layer j=n-1, according to equations (45), (46), (51),

[0207] w n =b n,1 (b n,2 +b n,3 -b n,4 w n+1 )=b n,1 (b n,2 +b n,3)=b n,2 (b n,2 +b n,3 )=b n-1,3 (b n-1,3 +b n,3 ) (53)

[0208]

[0209] According to the mathematical description of the circumferential strain rate of the inner wall of the nth and n-1th layers of the composite pipe described by equations (52) and (54), the circumferential strain rate of the inner wall of the first layer can be inferred:

[0210]

[0211] The correctness of the inference of formula (55) is proved below. When the 2nd to nth layers of the composite pipe are replaced by an equivalent j=2 layer, the circumferential strain rate of the inner wall of the equivalent layer is:

[0212]

[0213] b e2,3 =1(1b 2,3 +1b 3,3 +…+1b n-1,3 +1b n,3 )

[0214] Where: subscript "e" represents equivalent pipe parameters.

[0215] According to formula (45)(46)(51) and the equivalent layer outer wall And w3=0, we can get

[0216] w2=b 2,1 / (b 2,2 +b e2,3 -b e2,4 w3)=b 1,3 / (b 1,3 +b e2,3 )

[0217]

[0218] Since the above formula is exactly the same as formula (55), the conclusion is proved.

[0219] General water hammer velocity formula for thin-wall composite pipes:

[0220] Substituting formula (55) into formula (11), we can get:

[0221]

[0222] Substituting equation (56) into equation (12), the water hammer wave velocity of the thin-walled composite pipe can be obtained:

[0223]

[0224] Two special cases are analyzed below: thin-walled lining tunnel and underground composite pipe laying.

[0225] Calculation of water hammer wave velocity in thin-walled lining tunnels:

[0226] For thin-walled tunnels, formula (42)b should be used. n,3 Substitute b in formula (56) n,3 ,but:

[0227]

[0228] Substituting equation (58) into equation (12), the water hammer wave velocity of thin-walled lining tunnel can be obtained:

[0229]

[0230] Water hammer wave velocity of underground pipelines:

[0231] At present, a large number of water pipelines are laid underground and backfilled with soil (soil, sand and gravel) on top of the pipelines. When the backfill soil structure around the pipeline is loose and allows the pipeline to move axially, the effect of the backfill soil on the water hammer wave velocity can be ignored. When the backfill soil structure around the pipeline is dense and does not allow the pipeline to move axially, the effect of the backfill soil on the water hammer wave velocity needs to be considered. This situation is considered below.

[0232] The elastic coefficient and water hammer wave velocity of the inner wall of underground pipeline are:

[0233]

[0234] Where: E n is the elastic modulus of backfill soil, Pa; c s It is an empirical coefficient, which is related to the backfill thickness and Poisson's ratio. It is generally taken as c s =2.5. Analysis of the Kleck water hammer velocity formula: For thin-walled pipes, because:

[0235] ln(R j+1 / R j )=ln(1+e j / R j )≈e j / R j

[0236] Therefore, for thin-walled composite pipes, the elastic coefficient of the inner wall of the pipe in the improved Kleck water hammer velocity formula (3) can be rewritten as:

[0237]

[0238] or:

[0239]

[0240] The following application example will show that the Kleck water hammer wave velocity formula is only applicable to thin-walled composite pipes, but not to thick-walled composite pipes, and can be replaced by formula (59).

[0241] Step 5: Confirm the water hammer wave velocity calculation method and applicable scope:

[0242] Method 1: For composite pipes with thick wall layers, including linings, the general water hammer velocity formula (45)(48)(12) should be used to determine the water hammer velocity;

[0243] Method 2: For thin-walled composite pipes, the pipe elastic coefficient K can be calculated using equations (68), (70), and (74) according to the situation. c Then use formula (12) to calculate the water hammer wave velocity;

[0244] Method 3: Traditional calculation formulas for water hammer wave velocity in composite pipes, including the Coleck formula, Wylie and Streeter lined circular tunnel formula, are only applicable to thin-walled composite pipes, including thin-walled composite pipe lined tunnels, and are not applicable to composite pipes with thick wall layers.

[0245] The water hammer velocity of PCCP pipes is between 945 and 1183 m / s, and decreases with the increase of pipe diameter. This is quite different from the previous empirical value of 800 to 850 m / s. The main reason is that the empirical value is calculated according to the water hammer velocity formula for thin-walled pipes (32), which does not take into account the effect of the steel tube on the water hammer velocity.

[0246] When the pipeline is laid underground, the wave velocity in the pipeline increases slightly due to the influence of backfill (soil and gravel), but the impact is small.

[0247] Application examples:

[0248] Example 1: The tunnel section can be divided into 6 layers: the first layer is steel plate lining layer, the second layer is concrete lining layer, the third layer is reinforced concrete layer, the fourth layer is concrete lining layer, the fifth layer is cement grouting layer, and the sixth layer is surrounding rock layer. The inner wall radius R of each layer is j , wall thickness, elastic modulus E j , Poisson's ratio μ j , as listed in Table 1, where the ring reinforcement in the third layer of concrete is arranged at R3 = 3.2m, there are 4 steel bars with a diameter of d = 0.028m along the tunnel length L = 1m, and the equivalent steel layer thickness e3 = 4π(d2) 2 L=0.0025m; outer wall of surrounding rock layer R7→∞.

[0249] Table 1. Lined tunnel characteristic parameters and calculation results

[0250]

[0251]

[0252] For lined tunnels, the pipe support condition b can be used to select Calculate the water hammer wave velocity. The volume elastic modulus of the water body is K = 2.2E + 9Pa, and the density of water is ρ = 998.2kg / m 2 Three water hammer velocity calculation methods were used: Method 1 uses the general water hammer velocity formula for composite pipes (45)(48)(12), Method 2 uses the water hammer velocity formula for thin-walled lined tunnels (58)(59), and Method 3 uses the Kollek formula (62)(12). The calculation results are listed in Tables 1 and 2, respectively. Among the three water hammer velocity calculation methods, Method 1 fully considers the influence of the material mechanical properties and structural dimensions of the composite pipe, and the calculated water hammer velocity has the highest accuracy.

[0253] Table 2 lists the elastic coefficient K of the inner wall of the pipe c and the calculation results of water hammer wave speed a, where methods 2 and 3 calculate K c The values ​​are more than 1 times larger than those of method 1, with large deviations. As a result, the water hammer wave speeds calculated by the three methods are significantly different. The accurate water hammer wave speed calculated by method 1 is 1379m / s, and the approximate water hammer wave speeds calculated by methods 2 and 3 are 1442m / s and 1430m / s respectively.

[0254] Observe Table 1, K calculated by Method 2 c Component e j E j / R j c j and K calculated by method 3 c Component E j ln(R j+1 / R j ) / c j The cement grouting layer and surrounding rock layer account for the largest proportion, which can reach more than 90%. In other words, in the tunnel structure, since the cement grouting layer and surrounding rock layer are thick walls, they have the greatest impact on the water hammer wave velocity and play a dominant role. On the other hand, for the cement grouting layer, method 2 calculates K c The component is about 1.5 times that of method 3, and the difference between the two is obvious. However, in thin-walled layers such as concrete, steel bars, and steel plates, the K calculated by the two methods is c The difference in quantity is slight.

[0255] Table 2. Elastic coefficient K of the inner wall of the pipe c List of calculation results of water hammer wave velocity a

[0256]

[0257] Example 2: Lined prestressed concrete cylinder pipe (PCCPSL), which consists of a pipe core made of concrete and a steel cylinder, and a prestressed steel wire is wound around the outside of the steel cylinder, and then a pipe with a cement mortar protective layer is made. The pipe is connected with a socket joint. The minimum inner diameter of the PCCPSL pipe is 400mm, the minimum core thickness is 40mm, the minimum protective layer thickness is 25mm, and the steel cylinder thickness is 1.5mm [9] .

[0258] Since the condition of the prestressed hoop steel wire is unknown and it can be seen from Example 1 that the influence of the steel layer is small, it is not considered in the calculation. At this time, the pipeline can be divided into three layers: the first layer is the concrete lining, R1 = 0.2m, e1 = 0.0385m, E1 = 21×10 9 Pa, μ1 = 0.167; the second layer is a steel cylinder, the inner wall R2 = R1 + e1 = 0.2385m, e2 = 0.0015m, E2 = 200 × 10 9 Pa, μ2 = 0.3; the third cement mortar protective layer, inner wall R3 = R2 + e2 = 0.24m, e3 = 0.025m, E3 = 35 × 10 9 Pa, μ2=0.2, outer wall radius R4=R3+e3=0.265m.

[0259] For PCCP pipes, the water hammer wave velocity can be calculated according to the pipe support condition c, taking c j = 1. Method 2 uses the thin-wall water hammer velocity formula (56)(57). When methods 1, 2 and 3 are used to calculate the water hammer velocity respectively, the calculation results in Tables 3 and 4 can be obtained.

[0260] Observe Table 3, methods 2 and 3 calculate K c The values ​​are larger than those of method 1, with the maximum deviation reaching 17%. The water hammer wave velocities calculated by methods 1, 2, and 3 are 1184m / s, 1216m / s, and 1204m / s, respectively. c Component e j E j R j c j and K calculated by method 3 c Component E j ln(R j+1 / R j ) / c j The values ​​are close, among which the cement mortar layer and concrete lining account for a larger proportion, and the steel cylinder layer accounts for the smallest proportion. Since each layer of the PCCPSL pipe is thin-walled, the water hammer wave velocity calculated by methods 2 and 3 is closer to that of method 1.

[0261] Table 3. Elastic coefficient K of the inner wall of the pipe cList of calculation results of water hammer wave velocity a

[0262]

[0263] Table 4. PCCPSL pipe characteristic parameters and calculation results

[0264]

[0265] Example 3: Embedded prestressed concrete cylinder pipe (PCCPSE pipe), which consists of two layers of concrete inside and outside the steel cylinder to form the pipe core, and a circumferential prestressed steel wire is wound around the outside of the core concrete, and then a cement mortar protective layer is made. The pipe is connected by a socket joint. At present, the largest PCCP pipe in China has an inner diameter of 4000mm, a minimum core thickness of 260mm, a minimum protective layer thickness of 25mm, and a steel cylinder thickness of 1.5mm.

[0266] Since the condition of the prestressed hoop wire is unknown, it is not considered in the calculation. At this time, the PCCPSE pipeline can be divided into four layers: the first layer of concrete lining, inner wall R1 = 2.0m, e1 = 0.1m, E1 = 21×10 9 Pa, μ1 = 0.167; the second layer of steel cylinder, inner wall R2 = R1 + e1 = 2.1m, e2 = 0.0015m, E2 = 200 × 10 9 Pa, μ2 = 0.3; the third layer of concrete, inner wall R3 = R2 + e2 = 2.1015m, e3 = 0.16m, E3 = 21×10 9 Pa, μ3 = 0.167; the fourth cement mortar protective layer, inner wall R4 = R3 + e3 = 2.2615m, e4 = 0.025m, E3 = 35 × 10 9 Pa, μ2=0.2, outer wall radius R5=R4+e4=2.2865m.

[0267] Similar to Example 2, the water hammer wave velocity can be calculated according to the pipeline support condition C. When the water hammer wave velocity is calculated using methods 1, 2 and 3 respectively, the calculation results in Tables 5 and 6 can be obtained.

[0268] Observe Table 5, methods 1, 2, and 3 calculate K c The values ​​are close, with a deviation of 6%, making the water hammer velocity values ​​of the three methods very close, namely 945.5m / s, 961.5m / s, and 953.8m / s. c Component e j E j R j c j and K calculated by method 3 c Component E j ln(R j+1 R j )cj The values ​​are close, with the concrete lining accounting for a larger proportion and the steel cylinder layer accounting for the smallest proportion.

[0269] Table 5. Elastic coefficient K of the inner wall of the pipe c List of calculation results of water hammer wave velocity a

[0270]

[0271] Table 6. PCCPSE tube characteristic parameters and calculation results

[0272]

[0273] According to the calculation results of Examples 1, 2 and 3, an important conclusion can be obtained: the water hammer wave velocity calculation methods 2 and 3 are only applicable to thin-walled composite pipes, including thin-walled composite pipe-lined tunnels.

[0274] Since the PCCP pipes in Examples 2 and 3 are the smallest and largest diameter PCCP pipes that can be produced in my country, respectively, and the lining thickness increases with the increase of water pressure, resulting in an increase in the water hammer wave velocity, an important conclusion can be drawn from this: the water hammer wave velocity of the PCCP pipe decreases with the increase of the pipe diameter, and the value is between 945 and 1183 m / s. It should be noted that if the effect of the prestressed annular steel wire is considered, the water hammer wave velocity of the PCCP pipe will increase slightly. This is quite different from the previous empirical value of 800 to 850 m / s. The main reason for this large deviation is that the empirical value is calculated according to the water hammer wave velocity formula (32) for thin-walled pipes, without considering the influence of the steel cylinder on the water hammer wave velocity.

[0275] Example 4: The type and size of the pipeline are the same as those in Example 2. The pipeline is laid underground and the elastic modulus of the backfill soil is 5E+7. The water hammer wave velocity can be calculated according to the pipeline support condition b. Method 2 uses the thin-wall water hammer wave velocity formula (60)(61). When methods 1, 2 and 3 are used to calculate the water hammer wave velocity respectively, the calculation results in Table 7 can be obtained.

[0276] When using method 1, the water hammer wave velocity in Example 2 is 1183m / s, while in this example it is 1190m / s. It can be concluded that when the pipeline is laid underground, the pipeline wave velocity increases slightly due to the influence of backfill soil.

[0277] Table 7. Elastic coefficient K of the inner wall of the pipe c List of calculation results of water hammer wave velocity a

[0278]

[0279] Finally, it should be noted that the above is only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred arrangement scheme, a person skilled in the art should understand that the technical solution of the present invention (such as the form of the pipeline, the application of various formulas, the sequence of steps, etc.) can be modified or replaced by equivalents without departing from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for analyzing and calculating water hammer wave velocity of a composite pipeline, characterized in that: The steps of the method are as follows: Step 1, determining the functional relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipeline: according to the original definition of the water hammer wave velocity, determining the functional relationship between the water hammer wave velocity and the circumferential strain rate of the inner wall of the pipeline; The original definition of water hammer wave velocity a in water pipeline is: Where: K is the bulk elastic modulus of water; ρ is the liquid density; A is the rate of change of water pressure on the inner wall of the pipe; i is the water-passing cross-sectional area of ​​the inner wall of the pipe; is the change rate of the water-passing cross-sectional area of ​​the inner wall of the pipe; Relative expansion rate of the water-passing cross-sectional area of ​​the pipeline: Where: is the circumferential strain rate of the inner wall of the pipe; When defining: Where: K c is the elastic coefficient of the inner wall of the pipe in contact with water; Then K c The pipe water hammer wave velocity a0 is defined as: Step 2: Establishment of the basic equation for pipeline water hammer wave velocity: Establish the basic equation for the circumferential strain rate of a thick-walled cylinder and the circumferential, radial and axial stress change rates, and obtain the circumferential strain rate of the inner wall of the pipeline according to the pipeline support conditions. and the circumferential strain rate of the outer wall The basic equation is: Where: E is Young's elastic modulus; c1 is a constant related to the pipe support; R i is the radius of the inner wall of the pipe; R o is the radius of the pipe outer wall; p i is the inner wall pressure of the tube; p o is the outer wall pressure of the pipe; μ is the Poisson's ratio of the pipe material; For single-layer thick-walled pipes, Water hammer wave speed a1 formula: Step 3, accurate analysis of water hammer wave velocity of composite pipe: establish the basic equation of the relationship between the circumferential strain rate of the composite pipe wall in contact with water and the strain rates of other layers, interface pressure, pipe mechanical parameters and structural dimensions, and derive the general formula for accurately solving the water hammer wave velocity of composite pipe by analysis; The composite pipe is a pipe composed of two or more single-layer pipes tightly fitted together. The water hammer wave velocity a depends on the circumferential strain rate of the pipe wall in contact with water, that is, the circumferential strain rate of the inner wall of the first layer of the composite pipe. and It is not only related to the strain rate of the first layer's outer wall, but also to the strain rates of the inner and outer walls of other layers; Circumferential strain rate of the inner wall of layer j Where: c j is a constant related to the support of the j-th layer of pipeline, j is the layer number of the composite pipeline, j = n, n-1, n-2, ..., 1; n is the number of layers of the composite pipeline, which is a natural number; E j is the Young's elastic modulus of the j-th layer of pipe; E j 、E j+1 is the radius of the j-th layer pipeline and the j+1-th layer pipeline; μ j is the Poisson’s ratio of the j-th layer pipeline material; Circumferential strain rate of the outer wall of the jth layer At the interface between the j-1 layer outer wall and the j-th layer inner wall of the composite pipe, under the influence of water pressure, the j-1 layer outer wall will push the j-th layer inner wall outward, and the j-1 layer outer wall will produce a circumferential strain rate At the same time, the inner wall of the jth layer will push the outer wall of the j-1th layer inward, generating a circumferential strain rate As a result, the two layers are squeezed tightly together. At this time, the strain rate boundary condition at the interface between the two layers is: Where: Where: In the imperative form: During the pipe water hammer process, At this time, take: w n+1 ＝0 Where: w j 、w j+1 are the transition symbols of the jth layer and the j+1th layer respectively; b j,1 、b j,2 、b j,3 、b j,4 are coefficients related to the mechanical properties and structural parameters of each layer of the composite pipe material; Elastic coefficient K of the inner wall of the composite pipe in contact with water c0 : in: b is the circumferential strain rate of the first layer of pipe wall in contact with water; 1,3 、b 1,4 , w2 is the coefficient and transition symbol related to the mechanical properties and structural parameters of each layer of the composite pipe material; In summary, the calculation procedure for the water hammer wave velocity of the composite pipe is: algebraic recursion calculates w n 、w n-1 ,…,w2, and then calculate the elastic coefficient K of the composite tube c0 , and finally by the formula: Calculate the water hammer wave velocity; in the derivation process, the influence of the composite pipe on the circumferential strain rate is fully considered The various factors of the water hammer wave velocity calculated in this way are accurate solutions; Step 4, analysis of water hammer wave velocity of thin-walled composite pipes: Analyze various factors of the circumferential strain rate of the thin-walled composite pipe in contact with water, and propose an analytical formula for water hammer wave velocity, including analytical formulas for water hammer wave velocity of thin-walled lined tunnels and underground buried pipes; General elastic coefficient K of thin-wall composite pipe c1 : Water hammer wave velocity a1 of thin-wall composite pipe: Elastic coefficient K of thin-walled lining tunnel c2 : Water hammer wave velocity a2 in thin-walled lining tunnel: The elastic coefficient and water hammer wave velocity of the inner wall of underground pipeline are: Where: E n is the elastic modulus of backfill soil; c s is the empirical coefficient; Step 5, confirm the water hammer wave velocity calculation and applicable scope: Method 1: For composite pipes with thick wall layers, including linings, the general water hammer wave velocity formula for composite pipes is used: For composite pipes with thick wall layers, including linings, the following water hammer wave velocity formula is used to determine the water hammer wave velocity: Elastic coefficient K of thick-walled composite pipe c : Method 2: For thin-wall composite pipes, the water hammer velocity formula for thin-walled lined tunnels is used: Elastic coefficient K of thin-walled lining tunnel c2 : Water hammer wave velocity a2 in thin-walled lining tunnel:

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