CN103806478B - A kind of measuring method of Deformation of Diaphragm Wall - Google Patents

A kind of measuring method of Deformation of Diaphragm Wall Download PDF

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CN103806478B
CN103806478B CN201410018203.3A CN201410018203A CN103806478B CN 103806478 B CN103806478 B CN 103806478B CN 201410018203 A CN201410018203 A CN 201410018203A CN 103806478 B CN103806478 B CN 103806478B
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diaphragm wall
point
cloud
deformation
working base
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CN103806478A (en
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谢雄耀
卢晓智
朱侃唯
刘欢
王培�
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Tongji University
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Tongji University
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Abstract

The invention discloses a kind of measuring method of foundation ditch Deformation of Diaphragm Wall, in data acquisition, adopt three-dimensional laser scanner can Quick Measurement diaphragm wall three-dimensional coordinate, adopt working base point as splicing control point, Controling network is set up by laying deformation monitoring working base point and reference point, scanning work basic point while scanning diaphragm wall, can the point cloud registering of diaphragm wall in the absolute coordinate system of Controling network; In data processing, a some cloud rule mesh is adopted to format modeling technique, reconstruct diaphragm wall threedimensional model; The diaphragm wall threedimensional model of different times is compared, the distortion of all positions of diaphragm wall can be obtained.This measuring method can measure the bulk deformation of diaphragm wall in excavation process, without the need to laying monitoring point, reducing the interference to construction, measuring efficiency very high, for foundation pit construction provides comprehensive monitored data.

Description

A kind of measuring method of Deformation of Diaphragm Wall
Technical field
The invention belongs to base pit engineering monitoring technical field, relate to base pit engineering technology for deformation monitoring, especially using diaphragm wall as the technology for deformation monitoring of the base pit engineering of space enclosing structure.
Background technology
Along with the fast development of urban construction, urban land resource becomes more and more in short supply, and building land develops into trend to the underground space, and base pit engineering project gets more and more.Owing to containing many uncertain factors in engineering geologic investigation and design mistake, add that urban architecture is intensive, excavation of foundation pit can impact the stability of surrounding buildings, so become important step indispensable in base pit engineering to monitorings such as the space enclosing structure distortion caused in foundation pit construction engineering and surrounding enviroment impacts.
In the base pit engineering of soft clay area, city, diaphragm wall is the most widely used foundation pit enclosure structure form, and the distortion of monitoring diaphragm wall also becomes the important content of pit retaining monitoring.The method measuring Deformation of Diaphragm Wall is traditionally, lays several measuring points at diaphragm wall, measures its lateral deformation with inclinometer pipe.But this monitoring method can only obtain the deformation data of certain several section, the bulk deformation information of foundation ditch accurately cannot be reflected.
Summary of the invention
The object of the present invention is to provide a kind of measuring method of Deformation of Diaphragm Wall, the distortion of diaphragm wall can be measured quickly and efficiently, without the need to laying monitoring point, reduce the interference to construction, measure efficiency very high, and the deformation information of diaphragm wall entirety can be obtained, for foundation pit construction provides comprehensive monitored data.
For achieving the above object, solution of the present invention is:
A measuring method for Deformation of Diaphragm Wall, comprises the following steps:
(1) outside influence of foundation excavation region, lay at least 3 reference points, and before excavation, adopt control survey method to measure the three-dimensional coordinate of described reference point to set up absolute coordinate system; In influence of foundation excavation region, lay the working base point of at least 3 deformation monitorings, during each issue deformation observation, described working base point and described reference point are carried out translocation, to obtain the coordinate of described working base point in absolute coordinate system;
(2) the laying mode of point standing posture is adopted to lay multiple survey station to arrange three-dimensional laser scanner along the sideline of foundation ditch diaphragm wall, set up the reference frame determined by three-dimensional laser scanner, three-dimensional laser scanner is used to obtain the some cloud of described working base point and diaphragm wall, according to principle of coordinate transformation, using the three-dimensional coordinate of described working base point in absolute coordinate system as benchmark, the point cloud of described diaphragm wall is converted into absolute coordinate system from reference frame, and deletes the noise spot in absolute coordinate system except diaphragm wall point cloud;
(3) according to the outline line of foundation ditch plan view, using the some cloud of the diaphragm wall of straightway as a sector division out; And the diaphragm wall of each section point cloud is divided into several regions, the some cloud in each region is changed into a characteristic point, is used for representing the three-dimensional coordinate in this region, all characteristic points form the threedimensional model of diaphragm wall;
(4) compare the diaphragm wall threedimensional model of different times, then in threedimensional model, the displacement of character pair point is exactly the distortion of diaphragm wall in this position.
In described step (1), the layout points position of reference point is evenly distributed and covers whole influence of foundation excavation region; Described each working base point is at least arranged with two reference points and two 3 D laser scanning survey station intervisibilities.
In described step (1), outside influence of foundation excavation region, lay 4 prisms as reference point, in influence of foundation excavation region, lay 4 targets as working base point.
In described step (1), the scope in described influence of foundation excavation region depends on the character on stratum, excavation of foundation pit depth H, body of wall embedded depth, the lower sleeping soft layer degree of depth and excavation Construction of Supporting method.
Preferably, described influence of foundation excavation region is in the scope of 1 ~ 4H outside foundation ditch boundary line.
In described step (1), described control survey method is that vertical control survey, horizontal control survey or Three dimensions control are measured.
In described step (2), described multiple survey station covers the Zone Full of diaphragm wall, three-dimensional laser scanner laid by survey station, selected diaphragm wall and working base point carry out scanning survey as scanning target, each survey station obtains the some cloud of a part of diaphragm wall, until the scanning work of all survey stations all completes, uses working base point as splicing reference point, the point cloud of all survey stations is integrated in a unified reference frame, thus obtains the some cloud of complete diaphragm wall.
In described step (2), the three-dimensional coordinate of working base point in absolute coordinate system is imported in some cloud the poster processing soft, as first group of data, import in some cloud the poster processing soft as second group of data with reference to the working base point in coordinate system and diaphragm wall point cloud, the point cloud function in some cloud the poster processing soft is utilized to splice two groups of data, using working base point as splicing control point during splicing, the frame of reference using the coordinate system of first group of data as splicing, with the point cloud registering of diaphragm wall the most at last in absolute coordinate system;
Utilize the some cloud frame selection tool in some cloud the poster processing soft to choose and depart from diaphragm wall point far away, be regarded as noise spot and delete.
Described some cloud the poster processing soft is any one in Cylone, Polyworks, RealworksSurvey, Geomagic.
In described step (3), the point cloud of diaphragm wall is divided into the square net region of several 20cm × 20cm, a coordinate system is defined to the some cloud in each region, x-axis is made to be horizontal direction, z-axis is vertical direction, and y-axis is the normal direction of diaphragm wall, and the dimension of following provisions coordinate is m, if putting cloud quantity in region is n, some cloud coordinate is:
p i=(x i,y i,z i),i=1,2,Ln
The border in region is determined by xmin and zmin two parameters, meets following condition:
x min ≤ x i ≤ + 0.2 z min ≤ z i ≤ z min + 0.2 i = 1,2 , L n
The then characteristic point p in this region *=(x *, y *, z *) can be determined by following formula:
x * = x min + 0.1 y * = 1 n Σ i = 1 n y i z * = z min + 0.1
Owing to adopting such scheme, the invention has the beneficial effects as follows:
The present invention relates to in-site measurement and data processing two aspect:
(1) in-site measurement aspect: for obtaining foundation ditch bulk deformation information, the present invention adopts the three-dimensional coordinate of three-dimensional laser scanner measurement diaphragm wall, three-dimensional laser scanner adopts the layout scheme of point standing posture, namely multiple survey station is laid along margin of foundation pit, each survey station obtains the some cloud of a part of diaphragm wall, adopt working base point (target) as splicing reference point, the some cloud of all survey stations is integrated in a unified coordinate system, thus obtains the some cloud of complete diaphragm wall.
Point cloud in order to ensure the diaphragm wall obtained at different times is in same coordinate system, at least 3 reference points (prism) are laid outside influence of foundation excavation region, and before excavation, adopt the three-dimensional coordinate of control survey method datum mark to set up absolute coordinate system; In influence of foundation excavation region, lay the working base point of at least 3 deformation monitorings, during each issue deformation observation, described working base point and described reference point are carried out translocation, to obtain the coordinate of working base point in absolute coordinate system.
Because three-dimensional laser scanning technique is a kind of " planar survey " technology, during using diaphragm wall as scanning target, it can obtain the three-dimensional coordinate of target surface measuring point in laser measurement mode, measuring speed quickly, several ten thousand even hundreds of thousands measuring points (namely three-dimensional coordinate data is called " some cloud ") can be measured each second, in measuring process, some cloud can be automatically recorded in the storage system of three-dimensional laser scanner, these coordinate datas can be derived at data processing stage.
(2) object of data processing is by diaphragm wall point cloud registering in absolute coordinate system, and sets up the threedimensional model of diaphragm wall.For the data processing technical issues that need to address, the present invention proposes a kind of points cloud processing method, and step is as follows:
(1) adopt working base point (target) as splicing control point, diaphragm wall point cloud is spliced;
(2) using the three-dimensional coordinate of working base point in absolute coordinate system as benchmark, by diaphragm wall point cloud registering to absolute coordinate system;
(3) noise spot beyond diaphragm wall is deleted;
(4) according to the outline line of foundation ditch plan view, using the some cloud of the diaphragm wall of straightway as independently sector division out;
(5) a some cloud rule mesh is adopted to format modeling technique, the diaphragm wall of each section point cloud is divided into several regions, point cloud in each region is changed into a characteristic point, is used for representing the three-dimensional coordinate in this region, all characteristic points form the threedimensional model of diaphragm wall.
After in-site measurement and data place technical limit spacing diaphragm wall surface profile, the diaphragm wall threedimensional model of different times is compared, in threedimensional model, the change in location of character pair point is exactly the distortion of diaphragm wall in this position, finally can obtain the bulk deformation information of foundation ditch.
Accompanying drawing explanation
Fig. 1 is the workflow diagram of the measuring method of illustrated Deformation of Diaphragm Wall;
Fig. 2 be Deformation of Diaphragm Wall measuring method layout schematic diagram;
In figure: 1, diaphragm wall, 3, three-dimensional laser scanner survey station, 4, working base point, 5, reference point, 6, boundary line, excavation influence area.
Detailed description of the invention
Below in conjunction with accompanying drawing illustrated embodiment, the present invention is further illustrated.
The invention discloses a kind of measuring method of Deformation of Diaphragm Wall, relate to in-site measurement and data processing two aspect.In-site measurement aspect, the present invention is using diaphragm wall 1 as scanning target, three-dimensional laser scanning technique is adopted to measure, the three-dimensional coordinate that thousands of spacing are the measuring point of several millimeters can be obtained, measuring speed quickly, point cloud can be automatically recorded in the storage system of three-dimensional laser scanner, these coordinate datas can be derived at data processing stage.Diaphragm wall point cloud registering in absolute coordinate system, and is set up the threedimensional model of diaphragm wall 1 by data processing stage.After in-site measurement and data place technical limit spacing diaphragm wall surface profile, diaphragm wall 1 threedimensional model of different times is compared, in threedimensional model, the change in location of character pair point is exactly the distortion of diaphragm wall 1 in this position, finally can obtain the bulk deformation information of foundation ditch.
As shown in Figure 1, after determining the objective for implementation of the space enclosing structure of diaphragm wall 1 of a base pit engineering as diaphragm wall measuring method, the monitoring of its bulk deformation comprises the following steps:
(1) outside influence of foundation excavation region, namely boundary line, excavation influence area 6 is outer lays at least 3 reference points 5, and before excavation, adopts the three-dimensional coordinate of control survey method datum mark 5 to set up absolute coordinate system; The working base point 4 of at least 3 deformation monitorings is laid in boundary line, excavation influence area 6 in influence of foundation excavation region, during each issue deformation observation, working base point 4 and reference point 5 are carried out translocation, to obtain the coordinate of working base point 4 in absolute coordinate system.
General, the scope in influence of foundation excavation region depends on character, excavation of foundation pit depth H, body of wall embedded depth, the lower sleeping soft layer degree of depth on stratum and excavates Construction of Supporting method, in the present embodiment, influence of foundation excavation region is in the scope of 1 ~ 4H outside foundation ditch boundary line.
As shown in Figure 2, because reference point 5 is positioned at outside influence of foundation excavation region, its position can not change, before excavation, by the coordinate of control survey method determination reference point 5, at least three reference points 5 can form an absolute coordinate system, and now the three-dimensional coordinate of reference point 5 is known.Working base point 4 is positioned at influence of foundation excavation region, and its position can change in absolute coordinate system.During each issue deformation observation, translocation is carried out, the coordinate of surveying work basic point 4 to working base point 4 and reference point 5, just can determine the coordinate of working base point 4 in absolute coordinate system by the three-dimensional coordinate of known reference point 5.
Wherein, control survey method can be divided into two large classes, the first kind is vertical control survey and horizontal control survey, vertical control survey can adopt leveling measuring method, electromagnetic distance measurement Trigonometric Leveling, and horizontal control survey can adopt triangulateration, traverse, GPS to measure and the method such as triangulation survey, trilateration survey.Equations of The Second Kind control survey method is that Three dimensions control is measured, GPS can be used to measure and triangulateration, traverse, measurement of the level and electromagnetic distance measurement trigonometric heighting combined method.Translocation refers to the working base point 4 by the known reference point 5 surving coordinate the unknown of coordinate, and concrete measuring method is identical with control survey.
In step (1), by laying multiple prism as reference point 5 outside influence of foundation excavation region, the layout points position of reference point 5 is uniformly distributed and covers whole influence of foundation excavation region.Because each prism can determine the three-dimensional coordinate of a reference point 5, according to geometry principle, at least need three reference points 5 can form a coordinate system, therefore in the present embodiment, the quantity of prism can not be less than 3.During actual measurement, number of prisms should more than 4, and reason has two: remaining prism still can be relied on when 1. guaranteeing that part lens lost efficacy to form coordinate system; 2. unnecessary prism makes measurement have redundancy, is conducive to improving certainty of measurement.But, if number of prisms is too much, the workload of measurement can be increased, reduce measurement efficiency, too increase the cost of measurement.Based on above-mentioned consideration, in the present embodiment, 4 number of prisms are set altogether as reference point 5, but can take the circumstances into consideration to increase according to actual conditions.
In influence of foundation excavation region, the quantity of working base point 4 is quite a few in 3, and working base point 4 should be tried one's best near reference point and three-dimensional laser scanner survey station 3, so that measure simultaneously.The target at working base point 4 place and the prism of reference point 5 are without the need to one_to_one corresponding, effect due to working base point 4 is that the reference frame of the absolute coordinate system of reference point 5 and 3 D laser scanning is connected, according to the principle of coordinate transform, each working base point 4 at least will be arranged with two reference points 5 and two 3 D laser scanning survey station 3 intervisibilities.In the present embodiment, the layout with prism is considered, lays 4 targets as working base point 4 in influence of foundation excavation region.
(2) the laying mode of point standing posture is adopted to lay multiple survey station 3 to arrange three-dimensional laser scanner along the sideline of foundation ditch diaphragm wall 1, set up the reference frame determined by three-dimensional laser scanner, three-dimensional laser scanner is used to obtain the some cloud of described working base point 4 and diaphragm wall 1, according to principle of coordinate transformation, using the three-dimensional coordinate of working base point 4 in absolute coordinate system as benchmark, the point cloud of diaphragm wall 1 is converted into absolute coordinate system from reference frame, and deletes the noise spot in absolute coordinate system except diaphragm wall 1 cloud.
Absolute coordinate system is being set up according to the three-dimensional coordinate of reference point 5, and after determining the coordinate of multiple working base point 4 in absolute coordinate system, three-dimensional laser scanner is adopted to obtain the some cloud of working base point 4 coordinate and diaphragm wall 1 during following each issue deformation observation, the coordinate of working base point 4 and diaphragm wall 1 cloud is all in a reference frame determined by three-dimensional laser scanner, reference frame is specified by the built-in system of three-dimensional laser scanner, with the initial point that the position at three-dimensional laser scanner place is coordinate system, in horizontal plane with certain two mutually perpendicular direction for x-axis and y-axis, take vertical direction as z-axis.Because the coordinate of working base point 4 in absolute coordinate system and reference frame is all known, according to principle of coordinate transformation, the coordinate of working base point 4 is converted into absolute coordinate system from reference frame, and the some cloud of diaphragm wall 1 also can be converted into absolute coordinate system from reference frame in this process.Measure each time and all perform said process, the some cloud of diaphragm wall 1 just can be made to be in same absolute coordinate system.
In step (2), during each issue deformation observation, three-dimensional laser scanner laid by survey station 3, selected diaphragm wall 1 and working base point 4 carry out scanning survey as scanning target, each survey station 3 obtains the some cloud of a part of diaphragm wall 1, until the scanning at all survey stations 3 place all completes.When each survey station 3 scans, the requirement that 3 working base points 4 could meet splicing at least to be scanned.Meanwhile, sweep limits should cover diaphragm walls 1 whole in this survey station 3 visual line of sight.Certainly blocking due to actual sight line, each survey station 3 may can only scan the diaphragm wall 1 of a part, so need to arrange multiple survey station 3, just can scan the whole diaphragm wall of foundation ditch 1.
The three-dimensional coordinate of working base point 4 in absolute coordinate system is imported in some cloud the poster processing soft, as first group of data, import in some cloud the poster processing soft as second group of data with reference to the working base point 4 in coordinate system and diaphragm wall 1 cloud, the point cloud function in some cloud the poster processing soft is utilized to splice two groups of data, using working base point 4 as splicing control point during splicing, the frame of reference using the coordinate system of first group of data as splicing, with the point cloud registering of diaphragm wall 1 the most at last in absolute coordinate system; Then the noise spot in absolute coordinate system except diaphragm wall 1 cloud is deleted; In order to obtain accurate diaphragm wall 1 threedimensional model, improve the reliability of measurement result, the some cloud frame selection tool in the present embodiment utilization point cloud the poster processing soft is chosen and is departed from diaphragm wall 1 point far away, is regarded as noise spot and deletes.
Above-mentioned some cloud the poster processing soft is any one in Cylone, Polyworks, RealworksSurvey, Geomagic.
(3) according to the outline line of foundation ditch plan view, using the some cloud of the diaphragm wall 1 of straightway as a sector division out; And diaphragm wall 1 cloud of each section is divided into several regions, the some cloud in each region is changed into a characteristic point, is used for representing the three-dimensional coordinate in this region, all characteristic points form the threedimensional model of diaphragm wall 1.
In step (3), the point cloud of diaphragm wall 1 is divided into the square net region of several 20cm × 20cm, a coordinate system is defined to the some cloud in each region, x-axis is made to be horizontal direction, z-axis is vertical direction, and y-axis is the normal direction of diaphragm wall 1, and the dimension of following provisions coordinate is m, if putting cloud quantity in region is n, some cloud coordinate is:
p i=(x i,y i,z i),i=1,2,Ln
The border in region is determined by xmin and zmin two parameters, meets following condition:
x min ≤ x i ≤ + 0.2 z min ≤ z i ≤ z min + 0.2 i = 1,2 , L n
The then characteristic point p in this region *=(x *, y *, z *) can be determined by following formula:
x * = x min + 0.1 y * = 1 n Σ i = 1 n y i z * = z min + 0.1
If foundation ditch is rectangle as shown in drawings, then diaphragm wall 1 can divide into four planes, if foundation ditch is other shapes, is generally polygon, then diaphragm wall 1 will be divided into multiple plane
(4) compare diaphragm wall 1 threedimensional model of different times, then in threedimensional model, the displacement of character pair point is exactly the distortion of diaphragm wall 1 in this position.
Above-mentioned is can understand for ease of those skilled in the art and use the present invention to the description of embodiment.Person skilled in the art obviously easily can make various amendment to these embodiments, and General Principle described herein is applied in other embodiments and need not through performing creative labour.Therefore, the invention is not restricted to above-described embodiment, those skilled in the art, according to announcement of the present invention, do not depart from improvement that scope makes and amendment all should within protection scope of the present invention.

Claims (10)

1. a measuring method for Deformation of Diaphragm Wall, is characterized in that: comprise the following steps:
(1) outside influence of foundation excavation region, lay at least 3 reference points, and before excavation, adopt control survey method to measure the three-dimensional coordinate of described reference point to set up absolute coordinate system; In influence of foundation excavation region, lay the working base point of at least 3 deformation monitorings, during each issue deformation observation, described working base point and described reference point are carried out translocation, to obtain the coordinate of described working base point in absolute coordinate system;
(2) the laying mode of point standing posture is adopted to lay multiple survey station to arrange three-dimensional laser scanner along the sideline of foundation ditch diaphragm wall, set up the reference frame determined by three-dimensional laser scanner, three-dimensional laser scanner is used to obtain the some cloud of described working base point and diaphragm wall, according to principle of coordinate transformation, using the three-dimensional coordinate of described working base point in absolute coordinate system as benchmark, the point cloud of described diaphragm wall is converted into absolute coordinate system from reference frame, and deletes the noise spot in absolute coordinate system except diaphragm wall point cloud;
(3) according to the outline line of foundation ditch plan view, using the some cloud of the diaphragm wall of straightway as a sector division out; And the diaphragm wall of each section point cloud is divided into several regions, the some cloud in each region is changed into a characteristic point, is used for representing the three-dimensional coordinate in this region, all characteristic points form the threedimensional model of diaphragm wall;
(4) compare the diaphragm wall threedimensional model of different times, then in threedimensional model, the displacement of character pair point is exactly the distortion of diaphragm wall in this Feature point correspondence position.
2. the measuring method of Deformation of Diaphragm Wall according to claim 1, is characterized in that: in described step (1), and the layout points position of reference point is evenly distributed and covers whole influence of foundation excavation region; Each described working base point is arranged with at least two reference points and at least two 3 D laser scanning survey station intervisibilities.
3. the measuring method of Deformation of Diaphragm Wall according to claim 1 and 2, it is characterized in that: in described step (1), outside influence of foundation excavation region, lay 4 prisms as reference point, in influence of foundation excavation region, lay 4 targets as working base point.
4. the measuring method of Deformation of Diaphragm Wall according to claim 1, it is characterized in that: in described step (1), the scope in described influence of foundation excavation region depends on the character on stratum, excavation of foundation pit depth H, body of wall embedded depth, the lower sleeping soft layer degree of depth and excavation Construction of Supporting method.
5. the measuring method of the Deformation of Diaphragm Wall according to claim 1 or 4, is characterized in that: described influence of foundation excavation region is in the scope of 1 ~ 4H outside foundation ditch boundary line.
6. the measuring method of Deformation of Diaphragm Wall according to claim 1, is characterized in that: in described step (1), and described control survey method is that vertical control survey, horizontal control survey or Three dimensions control are measured.
7. the measuring method of Deformation of Diaphragm Wall according to claim 1, it is characterized in that: in described step (2), described multiple survey station covers the Zone Full of diaphragm wall, three-dimensional laser scanner laid by survey station, selected diaphragm wall and working base point carry out scanning survey as scanning target, each survey station obtains the some cloud of a part of diaphragm wall, until the scanning work of all survey stations all completes, use working base point as splicing control point, the point cloud of all survey stations is integrated in a unified reference frame, thus obtain the some cloud of complete diaphragm wall.
8. the measuring method of the Deformation of Diaphragm Wall according to claim 1 or 7, it is characterized in that: in described step (2), the three-dimensional coordinate of working base point in absolute coordinate system is imported in some cloud the poster processing soft, as first group of data, import in some cloud the poster processing soft as second group of data with reference to the working base point in coordinate system and diaphragm wall point cloud, the point cloud function in some cloud the poster processing soft is utilized to splice two groups of data, using working base point as splicing control point during splicing, the frame of reference using the coordinate system of first group of data as splicing, with the point cloud registering of diaphragm wall the most at last in absolute coordinate system,
Utilize the some cloud frame selection tool in some cloud the poster processing soft to choose and depart from diaphragm wall point far away, be regarded as noise spot and delete.
9. the measuring method of Deformation of Diaphragm Wall according to claim 8, is characterized in that: described some cloud the poster processing soft is any one in Cylone, Polyworks, RealworksSurvey, Geomagic.
10. the measuring method of Deformation of Diaphragm Wall according to claim 1, it is characterized in that: in described step (3), the point cloud of diaphragm wall is divided into the square net region of several 20cm × 20cm, define a coordinate system to the some cloud in each region, make x-axis be horizontal direction, z-axis is vertical direction, y-axis is the normal direction of diaphragm wall, the dimension of following provisions coordinate is m, if putting cloud quantity in region is n, some cloud coordinate is:
p i=(x i,y i,z i),i=1,2,…n
The border in region is determined by xmin and zmin two parameters, meets following condition:
x m i n ≤ x i ≤ x m i n + 0.2 z m i n ≤ z i ≤ z m i n + 0.2 , i = 1 , 2 , ... n
The then characteristic point p in this region *=(x *, y *, z *) can be determined by following formula:
x * = x m i n + 0.1 y * = 1 n Σ i = 1 n y i z * = z min + 0.1 .
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