Method for testing shearing performance of blade crown wear-resistant block with inclined meshing surface
By designing a shear performance detection method for the tilted engagement surface of the leaf crown wear-resistant block, using a universal tensile testing machine and image recognition technology, the problem of difficulty in evaluating the shear performance of the blade wear-resistant block in the prior art is solved, and the combination of butt welding is achieved Effective evaluation of strength and wear-resistant block strength guides welding process optimization and material selection, extending the service life of the engine.
Patent Information
- Application Number
- CN202411192073.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-08-28
- Publication Date
- 2024-12-13
AI Technical Summary
The prior art lacks special testing methods to evaluate the shear performance of the wear-resistant block on the tilted engagement surface on the turbine working blades of the aircraft engine or gas turbine, making it difficult to effectively evaluate the welding bond strength and the wear-resistant block itself.
A method for detecting the shear performance of the blade crown wear-resistant block with an inclination of the meshing surface was designed. The shear performance of the wear-resistant block is calculated by using a universal tensile testing machine and a shear dual pair of the blade wear-resistant block through tensile or compression test. , including welding, measuring, calculating shear strength, and using image recognition methods to measure the length of the cross-section arc after fracture, calculating the shear area and shear strength.
The bonding strength of the wear-resistant block after welding and the blade matrix and the strength of the wear-resistant block itself are effectively evaluated, and the welding process optimization and material selection are guided, so as to avoid the wear-resistant block breaking or falling off under load, extending the service life of the engine.
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Figure CN119124880A_ABST
Abstract
Description
Technical Field
[0001] The invention relates to a shear performance testing method for blade crown wear blocks with inclined meshing surfaces, belongs to the field of mechanical performance testing, and is mainly used for evaluating the welding bonding strength of blade crown wear layers of turbine working blades of aircraft engines or gas turbines. Background Art
[0002] The turbine blades of aircraft engines and gas turbines are mainly made of high-temperature alloy precision casting, and most of them are serrated crown structures. The structures of various types of blades are similar, and the blade crown model is similar to Figure 1 As shown. During the operation of aircraft engines and gas turbines, due to blade vibration or high temperature deformation, the blades are subjected to changing tangential forces, and adjacent blade crowns inevitably collide and rub. Once the blade crown is deformed, worn or broken due to collision, friction and variable load, the blade gap will be too large, resulting in excessive blade vibration, and the blade is prone to breakage, causing the engine or gas turbine to be scrapped. Therefore, adjacent blade crowns are generally separated by high-temperature wear-resistant blocks to reduce the failure of parts caused by blade crown wear.
[0003] At present, wear-resistant blocks are usually prepared by argon arc welding surfacing process. The welding wire used to prepare wear-resistant blocks is usually T800 or CM64 and other brands of wear-resistant welding wire. The wear-resistant layer material is brittle, high in hardness, and has high wear resistance. Therefore, the physical and mechanical properties of the same blade base material are quite different. At the same time, surfacing welding to prepare blade crown wear-resistant blocks is a subsequent process carried out on cast blades. It is not an integrated manufacturing process, so there will be an obvious interface effect between the wear-resistant layer and the blade crown. At the same time, the wear-resistant block itself is brittle and has low shear strength. For the above reasons, the bonding strength between the wear-resistant block and the substrate of the blade structure, as well as the strength of the blade wear-resistant block itself, need to be tested under simulated stress conditions.
[0004] At present, the shear strength of wear-resistant blocks prepared by wear-resistant welding wires is tested by welding sample joints. In fact, the structure and cooling effect of the blades during the welding process are quite different from those of the welded samples. At the same time, the bonding strength of the prepared wear-resistant blocks may also be different for different blade crown structures. In addition, the meshing surfaces of the wear-resistant blocks in some blades are inclined, that is, there are protrusions on the upper or lower surface of the blade crown, which also brings certain difficulties to the detection. Therefore, it is necessary to test the shear performance of wear-resistant blocks for blades with inclined meshing surface structures, but there is currently no dedicated testing method. This specification is designed to design a method for shear performance testing of blade crown wear-resistant blocks with inclined meshing surfaces. Summary of the invention
[0005] The present invention designs a method for detecting the shear performance of the wear-resistant layer of a blade crown with an inclined meshing surface. A universal tensile testing machine and a blade wear-resistant block shear pair are used to test and calculate the shear performance of the wear-resistant block in a tensile or compression test mode. The purpose is to evaluate the inherent strength of the wear-resistant blocks prepared by welding with wear-resistant welding wires of different types and manufacturers, as well as the bonding strength between the wear-resistant blocks and the blade substrate after welding.
[0006] The specific contents of the technical solution of the present invention are:
[0007] A method for detecting shear performance of a blade crown wear block with an inclined meshing surface comprises the following steps:
[0008] (1) The blades used for the shear performance test are finished blades. Wear-resistant blocks are welded on the meshing surfaces of the Z-shaped teeth of the adjacent blade crowns. After welding, they are machined into mutually matching inclined surfaces.
[0009] (2) Preparation of shearing pair: one end of the shearing pair has a thread that can be matched with the joint of the tensile testing machine. If the "tensile" shearing method is used, the other end is processed into a circular end 3 with a diameter of φ10 to φ12, and a step of 1.5 to 2.5 mm is formed on the upper surface 1 of the circular end 3 and the transition section 4; if the "compression" shearing method is used, the other end has a circular end 3 with a diameter of φ10 to φ12, and a step of 1.5 to 2.5 mm is left on the lower surface 2;
[0010] (3) Clamp the blade with a tool;
[0011] (4) Shear performance test of blade wear block: set the data acquisition frequency to 20 Hz. For the wear block with the inclined surface of the blade crown wear block facing upward, the stretching method is used for mobile loading; for the wear block with the inclined surface of the blade crown wear block facing downward, the compression method is used for mobile loading. When the prestressed load is applied, the load is applied at a constant displacement rate of 0.5 mm / min until damage occurs. When the force value gradually increases to the point where the wear block breaks, the testing machine automatically recognizes and stops the test. During the test, the force value versus displacement curve is recorded, and the maximum force value F is recorded.
[0012] (5) Dimension measurement after shearing and calculation of shearing area: After the specimen is damaged, when the damage mode is that the contact surface and the upper or lower surface of the blade crown are completely damaged, the length of the cross-sectional arc l1 after the contact surface is broken is measured by the image recognition method, and the length of the cross-sectional arc l2 of the contact surface is measured by the image recognition method. The cross-sectional shape is calculated according to the calculation formula of the trapezoid. The height of the trapezoid is the distance t0 from the contact surface to the upper surface of the blade crown or the length t0 of the fracture line from the contact surface to the meshing surface. The formula for calculating the shearing area is as follows:
[0013] A=0.5*(l1+l2)*t0
[0014] (6) Calculation of shear strength: Shear strength calculation formula:
[0015] τ=F / A.
[0016] The circular end is a semicircular or fan-shaped cylinder with any angle.
[0017] The shearing pair is made of the same alloy material as the blade base material.
[0018] The blade is clamped by a tooling method as follows: a hydraulic clamp is used below a universal tensile testing machine to fix a slide rail slot tooling that matches the shape of the blade tenon, and the threaded end of the upper shear pair is connected, and then the blade is installed in the slide rail slot tooling. During the process of sliding the blade in, the position of the lower slide rail slot tooling is adjusted to make the lower surface of the blade crown parallel to the bottom surface of the shear pair end; when the inclined surface of the blade crown wear-resistant block is upward, the upper surface of the end of the shear pair is tightly attached to the lower surface of the blade crown wear-resistant block, and an upward prestress is given by the testing machine to prevent slipping or offset; when the inclined surface of the blade crown wear-resistant block is downward, the lower surface of the end of the shear pair is tightly attached to the upper surface of the blade crown wear-resistant block, and a downward prestress is given by the testing machine to prevent slipping or offset.
[0019] The wear-resistant block is processed by argon arc welding with wear-resistant welding wire.
[0020] The material grade of the wear-resistant block is one of T800, CM64 and Setilite 6.
[0021] The material of the finished blade is one of single crystal alloy, directionally solidified alloy and equiaxed crystal.
[0022] The universal tensile testing machine is a microcomputer-controlled universal tensile testing machine; it meets 0.5-level calibration within the range of 1 to 100 KN, the acquisition frequency meets 10 Hz or more, the loading rate meets 0.1 mm / min or more, and the equipment has a hydraulic clamp.
[0023] The image recognition method for measuring the length of the cross-sectional arc after the contact surface is broken is as follows: taking a photo with a lens parallel to the sample measurement surface, importing the photo into the "CAXA" software, measuring the reference size, and extracting the required cross-sectional arc with a spline curve; by clicking the tool and querying the perimeter of the spline curve, the reference size and the data volume of the cross-sectional arc can be known, and the actual value of the reference size is substituted to calculate the length of the cross-sectional arc.
[0024] Beneficial effects of the present invention: The present invention can effectively judge the bonding strength between the blade wear block and the substrate after welding, and can also judge the strength of the wear block body after welding, and then judge whether the cladding process of the blade is suitable. In addition, this test method can also test the performance of different brands of welding wire materials after welding, and then guide the designer to select the appropriate welding wire material under a certain blade structure and a certain blade base material, and can also guide the welding process personnel to adjust the parameters to obtain the optimal bonding strength.
[0025] Understanding the bonding strength between the blade wear block and the base and its own strength is of great significance for the blade component level assessment. It can effectively evaluate the quality of the wear block before the blade is installed, and avoid the wear block from breaking or falling off under load, and then damaging other blades during engine operation, causing the entire engine to fail. BRIEF DESCRIPTION OF THE DRAWINGS
[0026] Figure 1 Schematic diagram of the location of the surfacing area and the size control of the wear-resistant layer in the prior art;
[0027] Figure 2 Schematic diagram of shear pair structure; a is a schematic diagram of shearing in the tensile mode of wear-resistant block, and b is a schematic diagram of shearing in the tensile and compression mode of wear-resistant block;
[0028] Figure 3 This is a schematic diagram of the structure of the slide rail slot tooling;
[0029] Figure 4 Schematic diagram of the shear test of the assembled blade;
[0030] Figure 5 Schematic diagram of the inclined blade crown meshing surface (i.e. the wear block meshing surface is not perpendicular to the upper and lower surfaces of the blade crown);
[0031] Figure 6 Schematic diagrams of two fracture modes; a is a side view of the wear-resistant block parallel to the blade crown direction; b is a top view of the upper surface of the blade crown; c is a bottom view of the lower surface of the blade crown.
[0032] Among them: 1-the surface matching with the blade wear block, 2-the lower surface of the end of the shear pair, 3-the semicircular end, 4-the middle transition section, 5-the upper surface of the blade crown wear block, 6-the meshing surface of the blade crown wear block, 7-the lower surface of the blade crown wear block DETAILED DESCRIPTION
[0033] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and examples:
[0034] A method for testing the shear performance of a blade crown wear layer with an inclined meshing surface, mainly for low-pressure turbine or power turbine working blades of aircraft engines and gas turbines, the blade material is a high-temperature alloy, such blades are sawtooth crown meshing, the meshing surface is prepared by argon arc welding with cemented carbide, and the meshing surface of the wear block is not perpendicular to the upper and lower surfaces of the blade crown, but at a certain angle. The shear performance testing method for such blade crown wear layer is characterized in that the steps of the method are:
[0035] (1) The blades for shear performance test should be finished blades, that is, the blade crown mating surface has completed welding and post-welding machining processes, the blade wear block is welded by wear-resistant welding wire through argon arc welding, the blade material is single crystal alloy, directionally solidified alloy, equiaxed crystal and other high-temperature alloys, and the applicable wear block material grades include T800, CM64, Setilite6, etc.
[0036] (2) Preparation of shear pair: One end of the shear pair has a thread that can match the joint of the tensile testing machine, and the other end is processed into a semicircular end 3 with a diameter of φ12. The diameter of the middle transition section 4 is φ8, that is, the width of the surface 1 that matches the blade wear block is 2 mm.
[0037] (3) Select a universal tensile testing machine. The equipment should be a microcomputer-controlled universal tensile testing machine, which should meet the 0.5 level calibration in the range of 1 to 100 KN, the acquisition frequency should be above 10 Hz, the loading rate should be above 0.1 mm / min, and the equipment should have a hydraulic clamp to facilitate the clamping of the blade fixing tooling.
[0038] (4) Clamp the blade with a fixture. Use a hydraulic clamp at the bottom of the universal tensile testing machine to fix the slide rail slot fixture that matches the shape of the blade tenon. Connect the threaded end of the upper shear pair, and then install the blade into the slide rail slot fixture. During the process of sliding the blade in, adjust the position of the lower slide rail slot fixture so that the lower surface of the blade crown is parallel to the bottom surface of the shear pair end. The meshing inclined surface of the wear block on the back side of the blade is upward, and the upper surface of the end of the shear pair is tightly attached to the lower surface 7 of the blade crown wear block, and the testing machine is used to give it an appropriate upward prestress (about 20N) to prevent slipping or offset; when the inclined surface of the blade crown wear block on the blade basin side is downward, the lower surface 2 of the end of the shear pair is tightly attached to the upper surface 5 of the blade crown wear block, and the testing machine is used to give it an appropriate downward prestress (about 80N) to prevent slipping or offset.
[0039] (5) Shear performance test of blade wear blocks: Set the data acquisition frequency to 20Hz. For the wear blocks on the back side of the blade (i.e., the meshing surface of the blade crown wear blocks is upward), the mobile loading is carried out by stretching; for the wear blocks on the blade basin side (i.e., the meshing surface of the blade crown wear blocks is downward), the mobile loading is carried out by compression. When the prestressed load is applied, the load is carried out at a constant displacement rate of 0.5mm / min until damage occurs. When the force value gradually increases to a certain value and the wear block breaks, the testing machine automatically recognizes and stops the test. During the test, the curve of the force value changing with the displacement is recorded, and the maximum force value F is recorded.
[0040] (6) Dimension measurement after shearing and calculation of shear area. After the specimen is damaged, the length of the arc line of the cross section after the contact surface is broken, l1, is measured by the image recognition method. The length of the arc line of the cross section of the contact surface, l2, is measured by the image recognition method. The cross-sectional shape is calculated according to the calculation formula of the trapezoid. The height of the trapezoid is the distance t0 from the contact surface to the upper surface of the blade crown (failure mode 1) or the length t0 from the contact surface to the fracture line of the meshing surface (failure mode 2). The measurement method of t0 under the two failure modes is detailed. The formula for calculating the shear area is as follows:
[0041] A=0.5*(l1+l2)*t0
[0042] When the failure mode is that the contact surface and the opposite side (the upper or lower surface of the blade crown wear block) are completely destroyed, the trapezoid height is the distance t0 from the contact surface to the opposite side. When the failure mode is that the contact surface and the wear block meshing surface are destroyed, and the opposite blade crown surface is not destroyed, measure the side fracture height (i.e. the distance between the bottom edge and the arc line of the contact surface section) t0.
[0043] The specific steps of the image recognition method are as follows: take a photo with the lens parallel to the sample measurement surface, import the photo into the "CAXA" software, measure the reference size, and use the spline curve to extract the required cross-sectional arc. By clicking the tool and querying the perimeter of the spline curve, the reference size and the data volume of the cross-sectional arc can be known. Substitute the actual value of the reference size to calculate the length of the cross-sectional arc.
[0044] (7,) Calculation of shear strength. Shear strength calculation formula:
[0045] τ=F / A
[0046] Example
[0047] (1) The first-stage working blade of a power turbine of a certain type of gas engine was selected. The wear block is about 6 mm long and 1.6 to 2.5 mm thick. The wear block on the blade basin side has an upper meshing surface that is inclined outward (i.e., the upper surface protrudes outward), and the wear block on the blade back side has an upper meshing surface that is inclined inward (i.e., the upper surface is concave inward and the lower surface protrudes outward). The blade base material is K438 equiaxed high-temperature alloy. The wear block is prepared by manual argon arc welding with T800 welding wire. After subsequent machining procedures, the shear strength test of the wear block is carried out.
[0048] (2) Preparation of shear pair. Generally, K438 alloy material, which is the same as the blade base material, is selected to prepare the blade shear pair. One end of the shear pair has a thread (size is M12) that can match the joint of the tensile testing machine, and the other end is processed into a semicircular end (diameter is φ12). The transition section in the middle is φ8, that is, the step size after processing is 2.
[0049] (3) Select a universal tensile testing machine. The equipment is a microcomputer-controlled universal tensile testing machine with a measuring range of 1 to 100 KN, a collection frequency of more than 10 Hz, a loading rate of more than 0.1 mm / min, and a hydraulic clamp on the upper and lower clamping ends of the equipment to clamp the blade fixing tooling.
[0050] (4) Clamp the blade with a fixture. Use a hydraulic clamp at the bottom of the universal tensile testing machine to fix the slide rail slot fixture that matches the shape of the blade tenon. Connect the threaded end of the upper shear pair. Then install the blade into the slide rail slot fixture. During the process of sliding the blade in, adjust the position of the lower slide rail slot fixture so that the lower surface of the blade crown is parallel to the bottom surface of the shear pair end. The meshing inclined surface of the wear block on the back side of the blade is upward, and the upper surface of the end of the shear pair is tightly attached to the lower surface 7 of the blade crown wear block, and the testing machine is used to give it an appropriate upward prestress (about 20N) to prevent slipping or offset; when the inclined surface of the blade crown wear block on the blade basin side is downward, the lower surface of the end of the shear pair is tightly attached to the upper surface of the blade crown wear block, and the testing machine is used to give it an appropriate (uncertain concept cannot be used, it can be a range) downward prestress (about 80N) to prevent slipping or offset.
[0051] (5) Shear performance test of blade wear blocks: set the data acquisition frequency to 20Hz, and use the stretching method to load the wear blocks on the back of the blade (i.e., the blade crown wear block meshing surface 6 is upward); use the compression method to load the wear blocks on the blade basin side (i.e., the blade crown wear block meshing surface 6 is downward); when the prestressed load is applied, the load is loaded at a constant displacement rate of 0.5mm / min until the damage occurs. When the force value gradually increases to a certain value and the wear block breaks, the testing machine automatically recognizes and stops the test. During the test, the force value versus displacement curve is recorded, and the maximum force value F is recorded.
[0052] (7) Dimension measurement after shearing and calculation of shear area. After the specimen is destroyed, the arc length l1 of the cross section after the contact surface is broken is measured by the image recognition method, and the arc length l2 of the cross section opposite to the contact surface is measured by the image recognition method. The cross-sectional shape is calculated according to the trapezoidal calculation formula, and the height is the side length from the contact surface to the opposite side or the upper surface of the leaf crown (i.e., t0). The formula for calculating the shear area is as follows:
[0053] A=0.5*(l1+l2)*t0
[0054] When the failure mode is that the contact surface and the opposite side (the upper or lower surface of the blade crown) are completely destroyed, the trapezoid height is the length from the contact surface to the opposite side (i.e. t0). When the failure mode is that the contact surface and the wear block meshing surface are destroyed, and the opposite blade crown surface is not destroyed, the side fracture height (the distance between the bottom edge and the arc line of the contact surface section) t0 is measured, and the cross-sectional shape is calculated according to the trapezoid calculation formula. The formula for calculating the shear area is as follows:
[0055] The specific steps of the image recognition method are as follows: take a photo with the lens parallel to the sample measurement surface, import the photo into the "CAXA" software, measure the reference size, and use the spline curve to extract the required cross-sectional arc. By clicking the tool and querying the perimeter of the spline curve, the reference size and the data volume of the cross-sectional arc can be known. Substitute the actual value of the reference size to calculate the length of the cross-sectional arc.
[0056] (8) Calculation of shear strength. Shear strength calculation formula:
[0057] τ=F / A
[0058] The following table shows an example of a blade with a back-side shear strength of 124.6 MPa and a base-side shear strength of 160.88 MPa.
[0059] Table 1. Test process data of the wear block on the back side of a test blade
[0060]
[0061]
[0062] Table 1. Test process data of the blade basin side wear block of a test blade
[0063]
[0064]
Claims
1. A method for testing the shear performance of blade crown wear blocks with inclined meshing surfaces. Features: The following steps are involved: (1) The blades for shear performance test are finished blades. Wear-resistant blocks are welded on the meshing surfaces of the Z-shaped teeth of the adjacent blade crowns. After welding, they are machined into mutually matching inclined surfaces. (2) Preparation of shear pair: One end of the shear pair has a thread that can match the joint of the tensile testing machine. If the "tensile" shearing method is used, the other end is processed into a round end with a diameter of φ10 to φ12, and a step of 1.5 to 2.5 mm is formed on the upper surface of the end and the transition section; if the "compression" shearing method is used, the other end is a round end with a diameter of φ10 to φ12 and a step of 1.5 to 2.5 mm is left on the lower surface; (3) Clamp the blade with a tool; (4) Shear performance test of blade wear block: set the data acquisition frequency to 20 Hz. For the wear block with the inclined surface of the blade crown wear block facing upward, the stretching method is used for mobile loading; for the wear block with the inclined surface of the blade crown wear block facing downward, the compression method is used for mobile loading. When the prestressed load is applied, the load is applied at a constant displacement rate of 0.5 mm / min until damage occurs. When the force value gradually increases to the point where the wear block breaks, the testing machine automatically recognizes and stops the test. During the test, the force value versus displacement curve is recorded, and the maximum force value F is recorded. (5) Dimension measurement after shearing and calculation of shearing area: After the specimen is damaged, when the damage mode is that the contact surface and the upper or lower surface of the blade crown are completely damaged, the length of the cross-sectional arc after the contact surface is broken is measured by image recognition method. 1 , the length of the arc of the cross section of the contact surface is measured by image recognition method 2 The cross-sectional shape is calculated according to the trapezoidal calculation formula. The height of the trapezoid is the distance t from the contact surface to the upper surface of the blade crown. 0 Or the length t of the fracture line from the contact surface to the meshing surface 0 , the formula for calculating the shear area is as follows: A=0.5*(l 1 +l 2 )*t 0 (6) Calculation of shear strength: Shear strength calculation formula: τ=F / A.
2. The method for detecting the shear performance of the wear block of the blade crown with inclined meshing surface according to claim 1, Features: The circular end is a semicircular or fan-shaped cylinder with any angle.
3. The method for detecting shear performance of blade crown wear blocks with inclined meshing surfaces according to claim 1, Features: The shearing pair is made of the same alloy material as the blade base material.
4. The method for detecting shear performance of blade crown wear blocks with inclined meshing surfaces according to claim 1, Features: The blade is clamped by a tooling method as follows: a hydraulic clamp is used below a universal tensile testing machine to fix a slide rail slot tooling that matches the shape of the blade tenon, and the threaded end of the upper shear pair is connected, and then the blade is installed in the slide rail slot tooling. During the process of sliding the blade in, the position of the lower slide rail slot tooling is adjusted to make the lower surface of the blade crown parallel to the bottom surface of the shear pair end; when the inclined surface of the blade crown wear-resistant block is upward, the upper surface of the end of the shear pair is tightly attached to the lower surface of the blade crown wear-resistant block, and an upward prestress is given by the testing machine to prevent slipping or offset; when the inclined surface of the blade crown wear-resistant block is downward, the lower surface of the end of the shear pair is tightly attached to the upper surface of the blade crown wear-resistant block, and a downward prestress is given by the testing machine to prevent slipping or offset.
5. The method for detecting shear performance of blade crown wear blocks with inclined meshing surfaces according to claim 1, Features: The wear-resistant block is made by argon arc welding with wear-resistant welding wire.
6. The method for detecting shear performance of blade crown wear blocks with inclined meshing surfaces according to claim 1, Features: The material grade of the wear-resistant block is one of T800, CM64 and Setilite 6.
7. The method for detecting shear performance of blade crown wear blocks with inclined meshing surfaces according to claim 1, Features: The material of the finished blade is one of single crystal alloy, directionally solidified alloy and equiaxed crystal.
8. The method for detecting shear performance of blade crown wear blocks with inclined meshing surfaces according to claim 1, Features: The universal tensile testing machine is a microcomputer-controlled universal tensile testing machine; it meets 0.5-level calibration within the range of 1 to 100 KN, the acquisition frequency meets 10 Hz or more, the loading rate meets 0.1 mm / min or more, and the equipment has a hydraulic clamp.
9. The method for detecting shear performance of blade crown wear blocks with inclined meshing surfaces according to claim 1, Features: The image recognition method for measuring the length of the cross-sectional arc after the contact surface is broken is as follows: the lens is parallel to the sample measurement surface to take a photo, the photo is imported into the "CAXA" software, the reference size is measured, and the required cross-sectional arc is extracted using a spline curve; by clicking the tool and querying the perimeter of the spline curve, the reference size and the data volume of the cross-sectional arc can be known, and the actual value of the reference size is substituted to calculate the length of the cross-sectional arc.
Citation Information
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