Foldable pipeline system, movable oil conveying and transferring platform and oil conveying and transferring method

Abstract

The invention belongs to the technical field of oil conveying and transferring, and particularly relates to a foldable pipeline system, a movable oil conveying and transferring platform and an oil conveying and transferring method.The foldable pipeline system comprises an arm frame and a pipeline, and the arm frame comprises n arm sections which are sequentially hinged end to end; the next arm section can be rotationally unfolded relative to the previous arm section through the arm section driving mechanism; the pipeline comprises (n + 1) pipe joints which are sequentially hinged end to end, the (n + 1) th pipe joint is a tail pipe, and the tail pipe can be unfolded relative to the nth pipe joint through a pipe joint driving mechanism; the other pipe joints are fixed to the arm joints in a one-to-one correspondence mode, and the hinge points of the pipe joints coincide with the hinge points of the arm joints. The device can be used on site after being unfolded, the problem that pipeline connection and sealing are not tight easily due to on-site building is solved, the using place of the carrying movable chassis is not limited, and through the length design of the rotary base and the pipeline, conveying and transferring of oil can be guaranteed within the determined working range.

Description

Technical Field

[0001] The present invention belongs to the technical field of oil transfer, and in particular relates to a foldable pipeline system, a mobile oil transfer platform and an oil transfer method. Background Art

[0002] With the continuous advancement of robotics technology, more and more fields are becoming automated and intelligent. Countries around the world are trying to develop automated oil transportation equipment and apply it to real-world scenarios of oil supply security. However, at present, whether it is a large span or a small span, oil transfer is mostly based on fixed pipelines, and there is little research on mobile transfer devices. Fixed pipelines need to be built on site, which has a long cycle and low efficiency. In the process of building pipelines, it is easy to have poor sealing and other situations, which may cause safety problems, consume manpower and material resources, and are difficult to reuse. Summary of the invention

[0003] The purpose of the present invention is to solve the problems existing in the above-mentioned prior art and to provide a foldable pipeline system, a mobile oil transfer platform and an oil transfer method, which solves the problem of loose pipeline connections during on-site construction and improves the distribution and transportation efficiency of oil.

[0004] The present invention is realized through the following technical scheme: a foldable pipeline system, including an arm frame and a pipeline, the arm frame includes n arm sections hinged end to end in sequence; the next arm section can be rotated and unfolded relative to the previous arm section through an arm section driving mechanism; the pipeline includes n+1 pipe sections hinged end to end in sequence, the n+1th pipe section is the end pipe, and the end pipe can be unfolded relative to the nth pipe section through a pipe section driving mechanism; the remaining pipe sections are fixed on each arm section one by one and the hinge points of the pipe sections coincide with the hinge points of the arm sections.

[0005] Furthermore, it also includes a swivel base, on which a column is fixedly connected, and the first arm section of the arm frame is hinged on the column and is rotated and unfolded relative to the column through an arm section driving mechanism.

[0006] Furthermore, the arm section driving mechanism and the tube section driving mechanism are both hydraulic telescopic rods.

[0007] Furthermore, the length of the pipe segment satisfies the following relationship:

[0008]

[0009] Where, L 末 Indicates the length of the end tube, L 总 It represents the total length of the 1st to nth pipe sections, working H represents the total height of the column and the rotating base, R represents the set working rotating radius, and A represents the maximum descent height of the end pipe that remains vertical under the working rotating radius R.

[0010] Furthermore, a length of no more than 5 cm is added to the theoretical length of the last tube as the actual length of the last tube, and a length of no more than 5 cm is added to the theoretical length of the first tube segment as the actual length of the first tube segment. The actual lengths of the remaining tube segments are equal to the corresponding theoretical lengths.

[0011] The present invention also provides a mobile oil transfer platform, which includes the foldable pipeline system of the present invention and is fixedly connected to the mobile chassis through the rotary base; an oil pump is installed at the lower end of the terminal pipe.

[0012] Furthermore, an automatic exhaust valve is installed at the upper end of the terminal pipe.

[0013] The present invention also provides an oil transfer method, which adopts the mobile oil transfer platform of the present invention and includes the following steps: unfolding the foldable pipeline system to a set working radius, while keeping the horizontal coordinate of the end pipe unchanged and the end pipe in a vertical state, step-by-step controlling the end pipe to descend to the bottom of the oil tank by adjusting the unfolding angle of the pipe section relative to the horizontal line, and then extracting oil from the oil tank or inputting oil into the oil tank by an oil pump.

[0014] Furthermore, one-click deployment to a set working radius comprises the following steps: deploying from the first pipe section to the last pipe section in sequence, triggering the deployment of the next pipe section each time the deployment reaches a trigger angle, and locking the pipe section each time the deployment reaches a locking angle; the locking angle and the trigger angle are determined as follows: determining the angle between the first pipe section and the horizontal line in the vertical state of the last pipe and the angles between the pipe sections according to the set working radius, and using these as the locking angles of the pipe sections during the one-click deployment process; and setting the trigger angle before locking for the 1st to nth pipe sections on the basis of the locking angle.

[0015] Furthermore, when there are 4 pipe sections, the constraints of the step control are as follows:

[0016]

[0017] Wherein, β1 represents the angle between the first pipe section and the horizontal line, β2 represents the angle between the second pipe section and the first pipe section, β3 represents the angle between the third pipe section and the second pipe section, β4 represents the angle between the end pipe and the third pipe section, x represents the horizontal coordinate of the lower end of the end pipe before stepping, x′ represents the horizontal coordinate of the lower end of the end pipe after stepping, Δy represents the step length of the end pipe; θ1, θ2, θ3 represent the expansion angles of the first pipe section, the second pipe section, the third pipe section and the horizontal line before stepping, respectively; θ1′, θ2′, θ3′ represent the expansion angles of the first pipe section, the second pipe section, the third pipe section and the horizontal line after stepping, respectively; θ4, θ4′ represent the angles between the end pipe and the third pipe section before and after stepping, respectively.

[0018] Compared with the prior art, the beneficial effects of the present invention include:

[0019] 1. The foldable pipeline system of the present invention can shrink and expand the pipeline through the combination of the arm and the pipeline, and can be used on site after expansion, thus solving the problem of loose pipeline connection during on-site construction.

[0020] 2. The mobile chassis is used to carry the entire foldable piping system and can be moved by a tractor. It can be used at any location and can be reused without disassembly.

[0021] 3. The length design of the rotary base and pipeline fully considers the oil transportation and transfer range, and can ensure the oil transportation and transfer within the determined working range;

[0022] 4. It has one-button deployment and one-piece recovery functions, which is easy to operate; after deployment, it can enter the stepping mode, the end pipe is always perpendicular to the ground, and the matching oil tank can be directly inserted without adjusting the end angle, which improves the efficiency of the transfer platform construction.

[0023] 5. The end pipe is equipped with an oil pump, which controls the forward or reverse rotation of the oil pump motor to achieve two-way transportation of oil.

[0024] 6. The automatic exhaust valve ensures that the air in the pipeline is discharged after the oil transportation operation is completed, ensuring that there is no residual oil in the pipeline, and stable operation can be achieved under the condition of end load. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 It is a schematic diagram of the folded state of the foldable piping system;

[0026] Figure 2 It is a schematic diagram of the folded state of the pipeline;

[0027] Figure 3 It is a schematic diagram of the unfolded state of the foldable piping system;

[0028] Figure 4 It is a schematic diagram of the working area range of the foldable piping system;

[0029] Figure 5 This is the finite element force analysis diagram of the boom;

[0030] Figure 6 It is a structural schematic diagram of each arm section of the boom;

[0031] Figure 7 This is a structural schematic diagram of a mobile oil transfer platform;

[0032] Figure 8 A control flow chart that can be expanded with one click;

[0033] Fig. 9 This is a schematic diagram of the working status of the mobile oil transfer platform;

[0034] Fig.10 This is a schematic diagram of the installation position of the automatic exhaust valve;

[0035] Fig.11 It is a schematic diagram of the expansion angle of the pipe segment when it moves in the plane coordinate system;

[0036] Fig.12 It is a schematic diagram of the pipe segment angle when the pipe segment moves in the plane coordinate system;

[0037] Fig.13 This is the schematic diagram of stepper control;

[0038] Fig.14 This is the oil transfer flow chart. DETAILED DESCRIPTION

[0039] The present invention is further described in detail below in conjunction with the accompanying drawings.

[0040] 1) Dimensional design of foldable piping system

[0041] A foldable pipeline system comprises an arm frame and a pipeline, wherein the arm frame comprises n arm sections hinged end to end in sequence; the next arm section can be rotated and unfolded relative to the previous arm section through an arm section driving mechanism; the pipeline comprises n+1 pipe sections hinged end to end in sequence, the n+1th pipe section is the end pipe, and the end pipe can be unfolded relative to the nth pipe section through the pipe section driving mechanism; the remaining pipe sections are fixed on each arm section in a one-to-one correspondence, and the hinge points of the pipe sections coincide with the hinge points of the arm sections.

[0042] Preferably, the arm section driving mechanism and the tube section driving mechanism are both hydraulic telescopic rods. The fixed end and the movable end of the arm section driving mechanism are respectively fixedly connected to the previous arm section and the next arm section; the fixed end and the movable end of the tube section driving mechanism are respectively fixedly connected to the nth tube section and the last tube.

[0043] The following description will be given using a piping system with four pipe sections as an example.

[0044] refer to Figure 1 and Figure 2 As shown, the foldable piping system includes a boom and a pipeline 7, the pipeline includes 4 pipe sections made of aluminum alloy pipes: a first pipe section 701, a second pipe section 702, a third pipe section 703, and a fourth pipe section 704, the boom includes a first arm section 4, a second arm section 5, and a third arm section 6; the first pipe section 701, the second pipe section 702, and the third pipe section 703 are respectively fixedly connected to the first arm section 4, the second arm section 5, and the third arm section 6 through mounting brackets (such as clamps), the pipe sections are connected through rotating joints 3, and the arm sections are connected and supported by hydraulic telescopic rods 9, which are driven by hydraulic cylinders 8 to ensure the strength between the arm sections.

[0045] The boom is supported and driven to rotate by a slewing base 1. A column 2 is fixedly connected to the slewing base, and the first arm section 4 of the boom is hinged on the column 2 and is unfolded relative to the column 2 through a hydraulic telescopic rod, and the fixed end and the moving end of the hydraulic telescopic rod are fixedly connected to the column 2 and the first arm section 4 respectively.

[0046] The foldable piping system of the present invention can be designed with boom length and arm section number according to different types of oil storage tanks. This specific implementation is described by taking the construction of a large-span oil transfer platform to achieve large-span oil transfer as an example.

[0047] refer to Figure 3 As shown, assuming that the number of pipe sections is n+1, the oil tank depth and the working turning radius are determined based on the working status analysis. The end pipe is always in a vertical state when working, and the end pipe needs to drop vertically by a depth greater than or equal to the oil tank depth.

[0048] Considering the ideal state, except for the end tube, the angles of the other arms are all 180°, and the theoretical length of the tube section satisfies the following relationship:

[0049]

[0050] Where, L 末 Indicates the theoretical length of the end tube, L 总 It represents the theoretical total length of the 1st to nth pipe sections, working H represents the total height of the column and the rotating base, R represents the set working rotating radius, and A represents the maximum descent height of the end pipe that remains vertical under the working rotating radius R.

[0051] Considering the influence of the turntable and folding size, the theoretical length of the last tube is increased by no more than 5cm as the actual length of the last tube, and the theoretical length of the first tube section is increased by no more than 5cm as the actual length of the first tube section. The actual lengths of the remaining tube sections are equal to the corresponding theoretical lengths. The first tube section and the last tube are appropriately enlarged in length to allow the last tube to fully penetrate the bottom of the matching oil tank.

[0052] refer to Figure 4 As shown in the figure, according to the working constraints of the end pipe and the dimensions of each arm length, the maximum working radius, the maximum working radius, and the minimum working radius can be calculated to determine the general working area. This enables the construction of a large-span oil transfer platform and completes the large-span oil transfer.

[0053] The maximum working radius is when L1, L2, and L3 are horizontally extended into a straight line and the end pipe is vertical to the ground.

[0054] The working radius after one-button expansion refers to the general working radius. The general working radius can be set by yourself, that is, as the set working radius, in normal application scenarios, the operator can switch the equipment from the folded state to the expanded state through simple operations, so as to start work immediately and improve work efficiency.

[0055] 2) Lightweight design of foldable piping system

[0056] First, the cross-sectional structure of the boom is confirmed according to the stress state of the boom. Through analysis, each boom is designed. At the same time, through finite element analysis, weight-reducing holes are opened in places where the stress strength is not high. The finite element stress analysis refers to Figure 5 shown.

[0057] The boom can be simplified into a slender cantilever beam, whose main load is its own weight. It requires the boom to be strong, rigid and light. Therefore, the structure of the boom is generally designed as a box beam made of four steel plates welded together. The material is high-strength fine-grained alloy structural steel. In order to make full use of the excellent mechanical properties of high-strength steel, the beam is designed as a gradient beam according to the principle that the stress at various locations on the beam tends to be consistent. Figure 6 As shown, taking into account the strength, rigidity, wind resistance and stability of the boom system, a three-arm structure is adopted. The first two booms adopt a box-type structure, and the third boom adopts a trough-type structure. The entire boom is light in weight.

[0058] A row of waist-shaped weight-reducing holes are separated on each boom section, which not only reduces the overall mass but also reduces the side windward area of ​​the overall structure, thereby improving the wind resistance. The shape of the waist-shaped weight-reducing holes can increase the structural strength of the object to a certain extent. By reducing the actual material usage of the object, the waist-shaped weight-reducing holes can provide sufficient strength and rigidity while reducing the overall weight. The design of the waist-shaped weight-reducing holes makes the weight more concentrated around the holes, thereby effectively reducing the center of the overall weight distribution of the object. This can improve the balance and stability of the object.

[0059] 3) Oil transfer platform

[0060] refer to Figure 7 As shown, a mobile oil transfer platform includes a foldable pipeline system 1B, which is fixedly connected to a mobile chassis 1A through a rotating base; an oil pump is installed at the lower end of the terminal pipe. It also includes an electrical control system 1C for controlling the expansion and contraction of the foldable pipeline system and oil delivery, and a hydraulic system 1D for supporting the expansion and contraction of the mechanical arm and the horizontal operation of the chassis.

[0061] The foldable piping system has a one-touch unfolding function, which can remotely control the robotic arm to unfold from the locked position to the end positioning condition with one click. The foldable piping system is a set of serial mechanisms. When working, the electro-hydraulic proportional valve controls the extension and contraction of each hydraulic telescopic rod to change the unfolding angle of each pipe section and the horizontal rotation angle between the arm and the vehicle body, so that the end pipeline moves smoothly to the oil outlet of the tank truck. When the one-touch unfolding button is clicked on the remote control, the system controls the four hydraulic telescopic rods to unfold each pipe section in turn until the end positioning condition is reached.

[0062] One-click expansion to a set working radius includes the following steps: sequentially expanding from the first pipe section to the last pipe section, triggering the next pipe section to expand each time the expansion reaches a trigger angle, and locking the pipe section each time the expansion reaches a locking angle; the locking angle and the trigger angle are determined as follows: the angle between the first pipe section and the horizontal line in the vertical state of the last pipe and the angles between the pipe sections are determined according to the set working radius, and used as the locking angles of the pipe sections during the one-click expansion process; and setting the trigger angle before locking for the 1st to nth pipe sections on the basis of the locking angle.

[0063] refer to Figure 8 As shown, the specific operation steps of one-key deployment are as follows: the first step is to confirm that the remaining oil has been drained and the device is unlocked; the second step is to use the hydraulic telescopic rod to drive the first pipe section to deploy; the third step is to drive the second pipe section to deploy when the first pipe section is deployed to a trigger angle of 50°, and lock the hydraulic telescopic rod of the first pipe section after the first pipe section reaches a locking angle of 60°; the fourth step is to open the driving hydraulic telescopic rod of the third pipe section when the second pipe section is deployed to a trigger angle of 160°, and drive the third pipe section to deploy; when the second pipe section reaches a locking angle of 175°, lock the hydraulic telescopic rod of the second pipe section; the fifth step is to open the driving hydraulic telescopic rod of the fourth pipe section when the third pipe section is deployed to a trigger angle of 150°, and drive the fourth pipe section to deploy; when the third pipe section reaches a locking angle of 155°, lock the hydraulic telescopic rod of the third pipe section; the sixth step is to lock the hydraulic telescopic rod of the fourth pipe section (last pipe) when the fourth pipe section (last pipe) is deployed to a locking angle of 60°.

[0064] A self-locking device is provided during the expansion and contraction process, and the pipeline is provided with a liquid sensing detector. When the residual oil is not completely drained, the one-button expansion and retraction function of the pipeline is locked.

[0065] refer to Fig. 9 As shown, an oil pump is installed at the end pipe, and the oil pump is preferably an electric submersible oil pump 10. The oil can be transferred through the pipeline system and the oil storage tank. The motor of the oil pump is provided with forward and reverse functions. When the end mechanism is inserted into the bottom of the oil tank, the oil pump is turned on. The oil can be pumped from the matching oil tank through the pipeline system to complete the large-span extraction of the oil; the motor is controlled to reverse and the oil source is connected to the pipeline system to realize the transfer and distribution of the oil.

[0066] refer to Fig.10As shown, an automatic exhaust valve 11 is provided above the end pipe of the boom pipeline system to achieve stable operation under the end load. When designing the entire oil pipeline, an automatic exhaust valve is provided above the end pipe to ensure that the air in the pipeline is exhausted after the oil transportation operation is completed, to ensure that no oil remains in the pipeline, and to achieve stable operation under the end load.

[0067] Automatic exhaust valves are part of the prior art. The working principle of an automatic exhaust valve is to sense the presence of gas through an internal float, spring or other mechanical device, and automatically open the valve to release the gas. When too much gas accumulates in a pipe or container, the automatic exhaust valve will open, allowing the gas to be discharged from the system to maintain the normal operating pressure of the system.

[0068] 4) Oil transfer method

[0069] After the foldable pipeline system is unfolded to the specified position, the stepping mode can be turned on to keep the end oil pipe in a vertical state at all times, thereby realizing the vertical lifting of the end pipe.

[0070] When the step mode is turned on, the pipeline system moves in a plane coordinate system during vertical landing. The geometric relationship is referenced Fig.11 and Fig.12 shown.

[0071] From the figure we can conclude that:

[0072] β1+β2+β3+β4=450° (1)

[0073] x=L1 cosθ1+L2cosθ2+L3 cosθ3 (2)

[0074] y1=L1sinθ1+L2sinθ2+L3sinθ3-L4 (3)

[0075] After stepping:

[0076] x′=L1 cosθ1′+L2cosθ2′+L3 cosθ′ (4)

[0077] y2=L1sinθ1′+L2sinθ2′+L3sinθ3′-L4 (5)

[0078] The step length is:

[0079] Δy=L1(sinθ1′-sinθ1)+L2(sinθ2′-sinθ2)+L3(sinθ3′-sinθ3) (6)

[0080] θ4=90°-θ3 (7)

[0081] refer to Fig.13As shown, when the end tube keeps vertical lifting, x remains unchanged. To ensure the dynamic balance and stability of the system, the adjustment range of each arm is adjusted according to the arm length ratio. The contact travel switch is used to determine whether it is fully deployed. After it is fully deployed, it will automatically stop, and the vertical height is minimized to reduce air resistance. The adjustment constraint conditions are as follows:

[0082]

[0083] Wherein, β1 represents the angle between the first pipe section and the horizontal line, β2 represents the angle between the second pipe section and the first pipe section, β3 represents the angle between the third pipe section and the second pipe section, β4 represents the angle between the end pipe and the third pipe section, x represents the horizontal coordinate of the lower end of the end pipe before stepping, x′ represents the horizontal coordinate of the lower end of the end pipe after stepping, Δy represents the step length of the end pipe; θ1, θ2, θ3 represent the expansion angles of the first pipe section, the second pipe section, the third pipe section and the horizontal line before stepping, respectively; θ1′, θ2′, θ3′ represent the expansion angles of the first pipe section, the second pipe section, the third pipe section and the horizontal line after stepping, respectively; θ4, θ4′ represent the angles between the end pipe and the third pipe section before and after stepping, respectively.

[0084] refer to Fig.14 As shown in the figure, the tail pipe one-key vertical landing control block diagram. After one-key deployment, if the transport vehicle is within the specified working range, the tail pipe is directly above the tank mouth. Manually judge whether the tail pipe enters the tank mouth and control the vertical lifting of the tail pipe. Since the tail pipe is slightly longer than the depth of the tank, it cannot continue to descend when it reaches the bottom of the tank.

[0085] In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "connected" and "connection" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0086] The above technical scheme is only a specific implementation method of the present invention. For those skilled in the art, it is easy to make various types of improvements or modifications based on the principles disclosed in the present invention, and it is not limited to the technical scheme described in the above specific embodiments of the present invention. Therefore, the above description is only preferred and does not have a restrictive meaning.

Claims

1. A foldable piping system, characterized in that: It comprises an arm frame and a pipeline, wherein the arm frame comprises n arm sections hinged end to end in sequence; the next arm section can be rotated and unfolded relative to the previous arm section through an arm section driving mechanism; the pipeline comprises n+1 pipe sections hinged end to end in sequence, the n+1th pipe section is the end pipe, and the end pipe can be unfolded relative to the nth pipe section through the pipe section driving mechanism; the remaining pipe sections are fixed on each arm section one by one, and the hinge points of the pipe sections coincide with the hinge points of the arm sections.

2. The foldable piping system according to claim 1, characterized in that: It also includes a swivel base, on which a column is fixedly connected, and the first arm section of the arm frame is hinged on the column and is rotated and unfolded relative to the column through an arm section driving mechanism.

3. The foldable piping system according to claim 2, characterized in that: The arm section driving mechanism and the tube section driving mechanism are both hydraulic telescopic rods.

4. The foldable piping system according to claim 2, characterized in that: The theoretical length of the pipe section satisfies the following relationship: Where, L 末 Indicates the theoretical length of the end tube, L 总 It represents the theoretical total length of the 1st to nth pipe sections, working H represents the total height of the column and the rotating base, R represents the set working rotating radius, and A represents the maximum descent height of the end pipe that remains vertical under the working rotating radius R.

5. The foldable piping system according to claim 4, characterized in that: Add a length of no more than 5 cm to the theoretical length of the last tube as the actual length of the last tube, add a length of no more than 5 cm to the theoretical length of the first tube section as the actual length of the first tube section, and the actual lengths of the remaining tube sections are equal to the corresponding theoretical lengths.

6. A mobile oil transfer platform, characterized in that: It comprises the foldable piping system as described in claim 5, and is fixedly connected to the mobile chassis through the rotating base; an oil pump is installed at the lower end of the terminal pipe.

7. The mobile oil transfer platform according to claim 6, characterized in that: An automatic exhaust valve is installed at the upper end of the terminal pipe.

8. A method for transferring oil, characterized in that: The mobile oil transfer platform as described in claim 6 is used, and includes the following steps: unfolding the foldable pipeline system to a set working radius, and while keeping the horizontal coordinate of the end pipe unchanged and the end pipe in a vertical state, step-by-step controlling the end pipe to descend to the bottom of the oil tank by adjusting the unfolding angle of the pipe section relative to the horizontal line, and then extracting oil from the oil tank or inputting oil into the oil tank by an oil pump.

9. The oil transfer method according to claim 8, characterized in that: One-click expansion to a set working radius includes the following steps: sequentially expanding from the first pipe section to the last pipe section, triggering the next pipe section to expand each time the expansion reaches a trigger angle, and locking the pipe section each time the expansion reaches a locking angle; the locking angle and the trigger angle are determined as follows: the angle between the first pipe section and the horizontal line in the vertical state of the last pipe and the angles between the pipe sections are determined according to the set working radius, and used as the locking angles of the pipe sections during the one-click expansion process; and setting the trigger angle before locking for the 1st to nth pipe sections on the basis of the locking angle.

10. The oil transfer method according to claim 8 or 9, characterized in that: When there are 4 pipe sections, the constraints of the step control are as follows: Wherein, β1 represents the angle between the first pipe section and the horizontal line, β2 represents the angle between the second pipe section and the first pipe section, β3 represents the angle between the third pipe section and the second pipe section, β4 represents the angle between the end pipe and the third pipe section, x represents the horizontal coordinate of the lower end of the end pipe before stepping, x′ represents the horizontal coordinate of the lower end of the end pipe after stepping, Δy represents the step length of the end pipe; θ1, θ2, θ3 represent the expansion angles of the first pipe section, the second pipe section, the third pipe section and the horizontal line before stepping, respectively; θ1′, θ2′, θ3′ represent the expansion angles of the first pipe section, the second pipe section, the third pipe section and the horizontal line after stepping, respectively; θ4, θ4′ represent the angles between the end pipe and the third pipe section before and after stepping, respectively.

Images