A method for calculating deformation of adjacent underground pipelines caused by foundation pit excavation considering damage and aging

Abstract

The application discloses a kind of considering breakage and aging's foundation pit excavation caused adjacent underground pipeline deformation calculation method, including establishing mechanical model;Foundation pit side wall deformation calculation prediction model;Additional load calculation;Elastic foundation beam solution.The application introduces image source method and combines Pasternak elastic foundation beam model, the settlement of adjacent underground pipeline caused by foundation pit excavation is obtained by the deformation of foundation pit enclosure structure, and the application considers aging factor by pipeline overall stiffness reduction, considers pipeline breakage factor by pipeline partial position stiffness reduction, and enriches the influence research of foundation pit excavation on adjacent underground pipeline deformation.The method can make full use of actual monitoring data of foundation pit engineering, and the deformation of adjacent underground pipeline caused by foundation pit excavation calculated is more reliable.

Description

Technical Field

[0001] This invention belongs to the field of underground engineering technology, specifically relating to a method for calculating the deformation of adjacent underground pipelines caused by foundation pit excavation considering damage and aging. Background Technology

[0002] The vigorous development of underground space in recent years has led to an increasing number of foundation pit projects in cities, raising widespread concerns about the impact of foundation pit excavation on the surrounding environment. Urban areas have a dense network of underground pipelines, including communication cables, power lines, water supply and drainage pipes, and gas pipelines; damage to these pipelines could have disastrous consequences. Therefore, it is necessary to study the deformation of underground pipelines adjacent to foundation pits.

[0003] Current domestic and international research on the deformation theory of underground pipelines near foundation pits includes several approaches. Li Dayong et al. used Winkler's elastic foundation beam theory to fit the horizontal and vertical displacement curves of the soil at the underground pipeline location in a parabolic form, establishing equations for the vertical and horizontal displacements of underground pipelines during excavation near foundation pits. Zhang Chenrong et al. proposed prediction curves for soil settlement outside the pit and horizontal displacement deformation of the retaining wall, obtaining the displacement and internal forces of underground pipelines caused by foundation pit excavation based on the displacement control method. Liu Hongyan et al., building on Li Dayong et al.'s work, used measured surface displacement to replace the fitted parabola, further verifying the correctness of the elastic foundation beam method for solving the vertical displacement of underground pipelines. Su Jun et al. proposed a pipeline displacement function for adjacent foundation pits based on monitoring data. However, when pipelines age or break, their stiffness is inevitably affected. Current research cannot yet consider the impact of underground pipeline damage and aging on pipeline deformation caused by foundation pit excavation.

[0004] In conclusion, considering the impact of pipeline aging and damage, it is necessary to propose a calculation method for the deformation of adjacent underground pipelines caused by foundation pit excavation. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for calculating the deformation of adjacent underground pipelines caused by foundation pit excavation.

[0006] This method for calculating the deformation of adjacent underground pipelines caused by foundation pit excavation includes the following steps:

[0007] Step 1: Establish a mechanical model

[0008] There is a rectangular foundation pit next to an existing underground pipeline, where L is the side length parallel to the pipeline axis (m); B is the side length perpendicular to the pipeline axis (m); d is the excavation depth of the foundation pit (m); H is the depth of the retaining structure (m); s is the distance from the pipeline axis to the side wall of the foundation pit (m); h is the pipeline burial depth (m); D is the outer diameter of the pipeline (m); with the center of the foundation pit excavation as the origin, x is the excavation width direction; y is the excavation length direction; and z is positive downwards.

[0009] Step 2: Establish a model for predicting the deformation of the foundation pit sidewalls

[0010] This invention establishes a deformation prediction model for foundation pit retaining structures applicable to internally braced support systems. A piecewise cosine function is used to fit the deformation increment of the retaining structure. The maximum deformation increment of the retaining structure is located near the excavation face, and the deformation increment v... i (η,d i )for:

[0011]

[0012] In the formula: v maxi d represents the maximum deformation of the retaining structure caused by the excavation of the i-th layer. i Let v be the excavation depth of the i-th layer. i (η,d i ) represents the deformation value of the retaining structure at depth η caused by the excavation of the i-th layer.

[0013] The ratio of the cumulative maximum deformation of the pit sidewall to the excavation depth, v max / d is used as the control parameter for sidewall deformation. If the deformation of the retaining structure meets the control parameter in each excavation, then the maximum sidewall deformation v caused by the excavation of the i-th layer is... maxi for:

[0014]

[0015] Plane strain ratio (PRS) quantitatively describes the spatial effects of the foundation pit:

[0016]

[0017] After considering spatial effects, the deformation increment θ of the retaining structure at any location on the sidewall during each excavation layer of the foundation pit i (λ,η,d i It can be easily estimated as follows:

[0018] θ i (λ,η,d i )=PSR(λ,d i )·v i (η,d i (4)

[0019] When the foundation pit is excavated to n layers, the depth of the bottom of the pit is d. i At that time, the displacement v(λ,η) of the sidewall is:

[0020]

[0021] The excavation depth of the foundation pit is the sum of the thickness of each excavation layer.

[0022] Step 3: Calculation of Additional Loads

[0023] According to the image source method, the displacement along the z-axis component S of a spherical gap with radius a at a point (B / 2, y, z) on the pipeline sidewall and at a point (x1, y1, z1) on the pipeline axis is known. z for:

[0024]

[0025] In the formula:

[0026] A horizontal displacement of magnitude v(|L / 2-y|,z) occurs at point (B / 2,y,z) on the sidewall. Differentiating the sidewall according to the equivalent volume principle yields:

[0027]

[0028] Substituting equation (7) into equation (6) and integrating along the depth direction of the enclosure structure, the vertical displacement S' generated by the deformation of the entire sidewall at point (x1, y1, z1) is obtained. z (y1):

[0029]

[0030] The vertical additional load F at point (x1, y1, z1) caused by the entire sidewall is then applied. z (y1):

[0031] F z (y1)=k·S' z (y1)·D (9)

[0032] In the formula: k is the soil subgrade coefficient, calculated using the Vesic formula. E0 is the elastic modulus of the soil, EI is the pipeline stiffness, and μ is the Poisson's ratio of the soil.

[0033] Step 4: Solving for the elastic foundation beam

[0034] A Pasternak foundation calculation model was established. Based on Pasternak's elastic foundation beam theory, the pipeline was simulated as an infinitely long beam resting on a series of springs, with a soil shear layer on the springs to account for soil continuity. Under the action of additional loads, the pipeline and the soil springs deformed.

[0035] The mechanical equations of the interaction between pipelines and the formation:

[0036]

[0037] In the formula: S(y1) is the vertical displacement of the pipeline at y1.

[0038] Equation (10) is solved using the finite difference method. Based on the pipeline discretization analysis, two virtual nodes are added at each end of the pipeline. The length of each element is l. Equation (10) is then written in the difference form:

[0039]

[0040] Where: G p For shear layer parameters, T is the shear layer thickness, T = 10D.

[0041] Based on the free conditions at both ends of the pipeline, after eliminating the displacement of the virtual nodes, equation (11) can be rewritten in matrix form:

[0042] [K t ][S]+[K s [S]-[G]=[F] (12)

[0043] In the formula: [K t [K] represents the pipeline displacement stiffness matrix; s [S] is the foundation stiffness matrix; [G] is the pile foundation horizontal displacement column vector; [F] is the soil shear stiffness matrix; [F] is the additional load column vector.

[0044]

[0045] [S] = [S0 S1 S2 ... S] n ] T (14)

[0046]

[0047]

[0048] [F] = [F0 F1 F2 ... F] n ] T (17)

[0049] Given the additional load, the displacement of underground pipelines caused by the excavation of the foundation pit can be obtained by combining equations (11) to (17).

[0050] Preferably, in step 4, when the underground pipeline is first laid, the stiffness of all points on the pipeline can be considered the same, which is the initial stiffness of the pipeline, i.e.:

[0051] E i I i =EI (18)

[0052] As pipelines age, aging and damage are inevitable. A pipeline strength reduction factor α, related to time t, is defined as 1 - t / 100 to account for pipeline aging. Therefore, after a service time t, the stiffness E at various points on the pipeline... i I i for:

[0053] E i I i =αEI (19)

[0054] Substituting equation (19) into the calculation step 4, we can obtain the solution for displacement of adjacent underground pipelines caused by foundation pit excavation considering pipeline aging.

[0055] When a pipeline experiences partial damage, the pipeline stiffness at the damaged location will also decrease. Let β be the ratio of the pipeline stiffness at the same location after damage to that before damage. Assume that the pipeline between nodes p and q experiences damage, and the stiffness E at each point on the pipeline at this time... i I i for:

[0056]

[0057] Substituting equation (20) into the calculation step 4, we can obtain the solution for displacement of adjacent underground pipelines caused by excavation of the foundation pit considering partial pipeline damage.

[0058] Based on the above technical solution, the beneficial effects of the present invention are:

[0059] (1) The present invention establishes a deformation prediction model for the foundation pit retaining structure. Based on the image source method, the horizontal displacement of the adjacent underground pipeline caused by the excavation of the foundation pit is obtained through the deformation of the foundation pit retaining structure. The deformation of the adjacent underground pipeline can be predicted before the foundation pit is excavated.

[0060] (2) The present invention takes into account the aging factor by reducing the overall stiffness of the pipeline and takes into account the pipeline damage factor by reducing the stiffness of a part of the pipeline. The calculated pipeline deformation is more in line with the actual engineering situation.

[0061] (3) This invention introduces the image source method and combines it with the Pasternak elastic foundation beam model. It obtains the settlement of adjacent underground pipelines caused by foundation pit excavation through the deformation of the foundation pit retaining structure. Furthermore, this invention considers aging factors by reducing the overall stiffness of the pipeline and pipeline damage factors by reducing the stiffness of certain parts of the pipeline, thus enriching the research on the impact of foundation pit excavation on the deformation of adjacent underground pipelines. This method can fully utilize the actual monitoring data of foundation pit projects, and the calculated deformation of adjacent underground pipelines caused by foundation pit excavation is more reliable. Attached Figure Description

[0062] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings:

[0063] Figure 1 The diagram shows the relative positions of the foundation pit and pipeline in an embodiment of the present invention. (a) is a top view, and (b) is a front view.

[0064] Figure 2 This is a schematic diagram of the deformation of the pit sidewall retaining structure in an embodiment of the present invention;

[0065] Figure 3 This is a diagram of the image source method calculation model in an embodiment of the present invention;

[0066] Figure 4 This is a calculation model diagram of the Pasternak foundation in an embodiment of the present invention;

[0067] Figure 5 This is a pipeline discrete analysis diagram in an embodiment of the present invention;

[0068] Figure 6 This is a comparison diagram of pipeline settlement and deformation calculations in Embodiment 1 of the present invention;

[0069] Figure 7 This is a comparison diagram of pipeline settlement and deformation calculations in Embodiment 2 of the present invention; Detailed Implementation

[0070] The present invention will be further described below with reference to embodiments. The description of the embodiments below is only for the purpose of helping to understand the present invention. It should be noted that those skilled in the art can make several improvements and modifications to the present invention without departing from the principle of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

[0071] This invention introduces the image source method and combines it with the Pasternak elastic foundation beam model. Based on the image source method, the horizontal displacement of adjacent underground pipelines caused by the excavation of the foundation pit is obtained through the deformation of the foundation pit retaining structure. Furthermore, this invention considers aging factors by reducing the overall stiffness of the pipeline and pipeline damage factors by reducing the stiffness of specific locations within the pipeline. Appropriate cases are selected for calculation, and the calculation model is as follows: Figure 1 As shown.

[0072] 1. Establish a mechanical model

[0073] In the diagram: There is a rectangular foundation pit next to an existing underground pipeline. L is the side length parallel to the pipeline axis (m); B is the side length perpendicular to the pipeline axis (m); d is the excavation depth (m); H is the depth of the retaining structure (m); s is the distance from the pipeline axis to the pit sidewall (m); h is the pipeline burial depth (m); D is the pipeline outer diameter (m); with the center of the foundation pit excavation as the origin, x represents the width of the excavation; y represents the length of the excavation; and z is positive downwards.

[0074] 2. Establish a model for predicting the deformation of the foundation pit sidewalls.

[0075] This invention establishes a deformation prediction model for foundation pit retaining structures applicable to internally braced support systems, such as... Figure 2 As shown, the deformation increment of the retaining structure is fitted using a piecewise cosine function. The maximum value of the deformation increment is located near the excavation face, and the deformation increment v i (η,d i )for:

[0076]

[0077] In the formula: v maxi d represents the maximum deformation of the retaining structure caused by the excavation of the i-th layer. i Let v be the excavation depth of the i-th layer. i (η,d i ) represents the deformation value of the retaining structure at depth η caused by the excavation of the i-th layer.

[0078] The ratio of the cumulative maximum deformation of the pit sidewall to the excavation depth, v max / d is used as the control parameter for sidewall deformation. If the deformation of the retaining structure meets the control parameter in each excavation, then the maximum sidewall deformation v caused by the excavation of the i-th layer is... maxi for:

[0079]

[0080] Plane strain ratio (PRS) quantitatively describes the spatial effects of the foundation pit:

[0081]

[0082] After considering spatial effects, the deformation increment θ of the retaining structure at any location on the sidewall during each excavation layer of the foundation pit i (λ,η,d i It can be easily estimated as follows:

[0083] θ i (λ,η,d i )=PSR(λ,d i )·v i (η,d i (4)

[0084] When the foundation pit is excavated to n layers, the depth of the bottom of the pit is d. i At that time, the displacement v(λ,η) of the sidewall is:

[0085]

[0086] The excavation depth of the foundation pit is the sum of the thickness of each excavation layer.

[0087] 3. Calculation of Additional Loads

[0088] like Figure 3 As shown, the image source method reveals that the displacement along the z-axis S produced by a spherical gap with radius a at a point (B / 2, y, z) on the pipeline sidewall at point (x1, y1, z1) on the pipeline axis is due to a spherical gap of radius a. z for:

[0089]

[0090] In the formula:

[0091] A horizontal displacement of magnitude v(|L / 2-y|,z) occurs at point (B / 2,y,z) on the sidewall. Differentiating the sidewall according to the equivalent volume principle yields:

[0092]

[0093] Substituting equation (7) into equation (6) and integrating along the depth direction of the enclosure structure, the vertical displacement S' generated by the deformation of the entire sidewall at point (x1, y1, z1) is obtained. z (y1):

[0094]

[0095] The vertical additional load F at point (x1, y1, z1) caused by the entire sidewall is then applied. z (y1):

[0096] F z (y1)=k·S' z (y1)·D (9)

[0097] In the formula: k is the soil subgrade coefficient, calculated using the Vesic formula. E0 is the elastic modulus of the soil, EI is the pipeline stiffness, and μ is the Poisson's ratio of the soil.

[0098] 4. Solving for beams on elastic foundations

[0099] like Figure 4 As shown, a Pasternak foundation calculation model is established. Based on Pasternak's elastic foundation beam theory, the pipeline is simulated as an infinitely long beam resting on a series of springs, with a soil shear layer on the springs to account for soil continuity. Under the action of additional loads, the pipeline and the soil springs deform.

[0100] The mechanical equations of the interaction between pipelines and the formation:

[0101]

[0102] In the formula: S(y1) is the vertical displacement of the pipeline at y1.

[0103] The solution is obtained by using finite difference pairs to solve equation (10), such as Figure 5 As shown, based on the pipeline discretization analysis, two virtual nodes are added at each end of the pipeline, and the length of each element is l. Equation (10) is then written in difference form:

[0104]

[0105] Where: G p For shear layer parameters, T is the shear layer thickness, T = 10D.

[0106] Based on the free conditions at both ends of the pipeline, after eliminating the displacement of the virtual nodes, equation (11) can be rewritten in matrix form:

[0107] [K t ][S]+[K s [S]-[G]=[F] (12)

[0108] In the formula: [K t [K] represents the pipeline displacement stiffness matrix; s [S] is the foundation stiffness matrix; [G] is the pile foundation horizontal displacement column vector; [F] is the soil shear stiffness matrix; [F] is the additional load column vector.

[0109]

[0110] [S] = [S0 S1 S2 ... S] n ] T (14)

[0111]

[0112]

[0113] [F] = [F0 F1 F2 ... F] n ] T (17)

[0114] Given the additional load, the displacement of underground pipelines caused by the excavation of the foundation pit can be obtained by combining equations (11) to (17).

[0115] In step 4, when the underground pipeline is first laid, the stiffness of all points on the pipeline can be considered the same, which is the initial stiffness of the pipeline, i.e.:

[0116] E i I i =EI (18)

[0117] As pipelines age, aging and damage are inevitable. A pipeline strength reduction factor α, related to time t, is defined as 1 - t / 100 to account for pipeline aging. Therefore, after a service time t, the stiffness E at various points on the pipeline... i I i for:

[0118] E i I i =αEI (19)

[0119] Substituting equation (19) into the calculation step 4, we can obtain the solution for displacement of adjacent underground pipelines caused by foundation pit excavation considering pipeline aging.

[0120] When a pipeline experiences partial damage, the pipeline stiffness at the damaged location will also decrease. Let β be the ratio of the pipeline stiffness at the same location after damage to that before damage. Assume that the pipeline between nodes p and q experiences damage, and the stiffness E at each point on the pipeline at this time... i I i for:

[0121]

[0122] Substituting equation (20) into the calculation step 4, we can obtain the solution for displacement of adjacent underground pipelines caused by excavation of the foundation pit considering partial pipeline damage.

[0123] Example 1:

[0124] Specific working conditions and parameters: The excavation dimensions of the JN station foundation pit are 240m × 21.3m, with a depth of 19.5m. The support structure adopts a diaphragm wall combined with an internal bracing system. A DN1200 drainage pipe, numbered P3, with an outer diameter of 1246mm and a wall thickness of 23mm, is installed parallel to the length of the foundation pit. It is made of concrete, buried at a depth of 6.71m, with a net distance of 4.89m from the foundation pit. E = 2.85 × 10⁻⁶. 4 MPa. The soil's elastic modulus is 80 MPa, and Poisson's ratio is 0.26. In the example, v max / d=0.075%, pipeline age t=0, no damage.

[0125] Under this operating condition, the calculated and measured values ​​of pipeline settlement are compared, as follows: Figure 6 As shown in the figure, the calculated values ​​of the method of the present invention are in good agreement with the measured values, verifying the reliability of the calculation method of the present invention. The measured data show that the maximum settlement of the pipeline caused by the excavation of the foundation pit occurs within the excavation area, with a maximum value of 6.9 mm. The pipeline settlement value is larger near the center of the foundation pit excavation and decreases towards both sides. The influence range of the pipeline settlement is approximately 150 m on each side.

[0126] Example 2:

[0127] Specific working conditions and parameters: The plan dimensions of a foundation pit in Pudong New Area are: rectangular, 101.5m × 56.4m, with the northeast corner cut off (35.3m × 21.4m), and a depth of 22.0m. A steel pipe, 4mm thick and 300mm in diameter, is buried 1m deep and approximately 5.87m from the edge of the pit. The soil compression modulus is 6.1MPa, and Poisson's ratio is 0.33. In the calculation example, v... max / d=0.28%, pipeline service life t=0, no damage.

[0128] Under this working condition, the settlement calculation value of the method of the present invention is as follows: Figure 7 As shown. The calculated values ​​by the method of this invention are in good agreement with the measured values. Compared with Example 1, the pipeline settlement range is correspondingly smaller in Case 2 because the excavation area of ​​the foundation pit is smaller.

[0129] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the disclosure herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this application are indicated by the following claims.

[0130] It should be understood that this application is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of this application is limited only by the appended claims.

Claims

1. A method for calculating the deformation of adjacent underground pipelines caused by foundation pit excavation, characterized in that, Includes the following steps: Step 1: Establish a mechanical model: There is a rectangular foundation pit next to the existing underground pipeline, where L is the side length in the direction parallel to the pipeline axis (in meters), and B is the side length in the direction perpendicular to the pipeline axis (in meters). d represents the excavation depth of the foundation pit, in meters (m). H represents the depth of the building envelope, in meters (m). s is the distance from the pipeline axis to the side wall of the pit, in meters; h is the pipeline burial depth, in meters; D is the pipeline outer diameter, in meters; with the center of the pit excavation as the origin, x is the direction of the pit excavation width; y is the direction of the pit excavation length; z is positive downwards. Step 2: Establish a model for predicting the deformation of the foundation pit sidewalls The deformation increment of the enclosure structure is fitted using a piecewise cosine function. for: (1) In the formula: v maxi Let d be the maximum deformation value of the retaining structure caused by the excavation of the i-th layer. i Let v be the excavation depth of the i-th layer. i (η,d i ) represents the deformation value of the retaining structure at depth η caused by the excavation of the i-th layer; The ratio of the cumulative maximum deformation of the pit sidewall to the excavation depth, v max / d is used as the control parameter for sidewall deformation. If the deformation of the retaining structure meets the control parameter in each excavation, then the maximum sidewall deformation caused by the excavation of the i-th layer is... for: (2) Plane strain ratio (PRS) quantitatively describes the spatial effects of the foundation pit: (3) After considering spatial effects, the deformation increment of the retaining structure at any location on the sidewall during each layer of excavation of the foundation pit A simplified estimate is as follows: (4) When the foundation pit is excavated to n layers, the depth of the bottom of the pit is d. i When, the displacement of the sidewall for: (5) Step 3: Calculation of Additional Loads The image source method can be used to identify a point on the side wall of a nearby pipeline. A point on the pipeline axis of a spherical gap with radius a The displacement generated at that location along the z-axis component for: （6） In the formula: ; Side wall point The size of the occurrence is The horizontal displacement, obtained by differentiating the sidewall according to the equivalent volume principle, is: (7) Substituting equation (7) into equation (6) and integrating along the depth direction of the enclosure structure, the deformation of the entire sidewall at point... The resulting vertical displacement : (8) Then the entire sidewall causes points Vertical additional load : (9) In the formula: k is the soil subgrade coefficient, calculated using the Vesic formula. E0 is the elastic modulus of the soil, EI is the pipeline stiffness, and μ is the Poisson's ratio of the soil. Step 4: Solving for the elastic foundation beam: A Pasternak foundation calculation model was established. Based on Pasternak's elastic foundation beam theory, the pipeline was simulated as an infinitely long beam resting on a series of springs, with a soil shear layer on the springs to consider soil continuity. Under the action of additional load, the pipeline and the soil springs deformed, and the displacement of the underground pipeline was obtained through calculation. The displacement of underground pipelines was calculated, specifically including: The mechanical equations of the interaction between pipelines and the formation: (10) In the formula: S(y1) is the vertical displacement of the pipeline at y1; Equation (10) is solved using the finite difference method. Based on the pipeline discretization analysis, two virtual nodes are added at each end of the pipeline. The length of each element is l. Equation (10) is then written in the difference form: (11) Where: G p For shear layer parameters, T is the shear layer thickness, T=10D; Based on the free conditions at both ends of the pipeline, after eliminating the displacement of the virtual nodes, equation (11) can be rewritten in matrix form: (12) In the formula: [K t [K] represents the pipeline displacement stiffness matrix; s [S] is the foundation stiffness matrix; [G] is the column vector of pile foundation horizontal displacement; [F] is the soil shear stiffness matrix; [G] is the column vector of additional load. (13) (14) (15) (16) (17) Given the additional load, the displacement of underground pipelines caused by the excavation of the foundation pit can be obtained by combining equations (11) to (17). In formula (10) Determined in the following ways: When underground pipelines are first laid, the stiffness of all points along the pipeline is assumed to be the same, which is the initial stiffness of the pipeline, i.e.: (18) As pipelines age, aging and damage inevitably occur. A pipeline strength reduction factor α = 1 - t / 100 is defined as a factor related to time t to account for pipeline aging. Therefore, after a service time t, the stiffness of each point on the pipeline... for: (19) Substituting equation (19) into step 4 of the calculation process, we obtain the solution for displacement of adjacent underground pipelines caused by excavation of the foundation pit, taking into account pipeline aging. When a pipeline experiences partial damage, the pipeline stiffness at the damaged location will also decrease. Let β be the ratio of the pipeline stiffness at the same location after damage to that before damage. When damage occurs between pipeline nodes p and q, the stiffness at each point on the pipeline will... for: (20) Substituting equation (20) into step 4 of the calculation process, we obtain the solution for displacement of adjacent underground pipelines caused by excavation of the foundation pit, considering partial pipeline damage.

Images