Column longitudinal steel bar blanking length calculation method

Abstract

The invention discloses a column longitudinal reinforcement blanking length calculation method, which comprises the following steps of: respectively calculating the lengths of longitudinal reinforcements of longitudinal columns on a bottom layer and a top layer, and calculating the longitudinal reinforcements of longitudinal columns on a middle layer from bottom to top layer by layer, so that the splicing positions of the longitudinal reinforcements are in a height range of a connectable area; and the height difference between the high-pile long steel bars and the low-pile long steel bars meets the requirement. And when the Markov state transition optimization method is adopted to optimize the connection mode of the middle-layer longitudinal steel bars, the optimal longitudinal steel bar splicing scheme and the optimal middle-layer longitudinal steel bar blanking list can be obtained. In addition, for the bottom-layer longitudinal steel bars and the top-layer longitudinal steel bars, the cutting scheme is optimized by adopting a genetic algorithm, and a raw material steel bar cutting scheme with the minimum waste rate can be obtained. According to the method, material breaking and sample turning optimization work of the longitudinal steel bars of the bottom layer, the middle layer and the top layer is comprehensively considered, the sample turning efficiency of the longitudinal steel bars is improved, and materials are saved while compliance and constructability are met.

Description

Technical Field

[0001] The invention relates to a method for calculating the cutting length of longitudinal steel bars of a column, and belongs to the field of intelligent construction. Background Art

[0002] During the construction of concrete structural columns, it is necessary to determine the length and connection method of the column reinforcement according to the design drawings and complete the reinforcement pattern. Since the longitudinal reinforcement of the column is usually thicker in diameter, heavier in weight and more expensive, it is necessary to comprehensively consider multiple factors such as constructability, material saving and compliance. Therefore, the pattern-making problem of the longitudinal reinforcement of the column has always been the focus and difficulty in the construction process. During the pattern-making process, the large-diameter longitudinal reinforcement of the column should be reasonably located and disconnected at each layer and connected through sleeves to ensure the convenience of construction; at the same time, the steel bars used should minimize waste to improve economy; the longitudinal reinforcement of the column is usually divided into two categories: low pile length and high pile length. The length of the connection area is staggered reasonably to meet the specification requirements of only 50% connection in the same connection area.

[0003] At present, the distance between the breakpoint of the longitudinal reinforcement of the column and the floor height of each floor is determined by on-site technicians based on experience. Due to the changes in factors such as the floor height and beam height of different floors, the diameter and number of longitudinal reinforcements, it is often difficult for technicians to fully take into account all aspects of the constraints, and it is easy to have problems such as setting the reinforcement connection area within the beam height range. Therefore, multiple calculations and reviews are required, which leads to inefficient sample-making process and suboptimal results, increasing the difficulty of construction and causing material waste. Summary of the invention

[0004] In view of the above-mentioned problems existing in the prior art of making the longitudinal reinforcement samples of columns, the present invention provides a method for calculating the cutting length of the longitudinal reinforcement samples of columns, which improves the efficiency of making the reinforcement samples, ensures compliance and constructability, and reduces material consumption.

[0005] In order to solve the above technical problems, the present invention includes the following technical solutions:

[0006] A method for calculating the cutting length of longitudinal steel bars of a column, wherein a building comprises N floors, a beam intersecting with a longitudinal column is arranged at the bottom of each floor, a roof beam is arranged at the roof of the building, and the longitudinal steel bars of the longitudinal column comprise long steel bars of a high pile and long steel bars of a low pile; the calculation method comprises the following steps:

[0007] Step 1: determine the length, diameter, and quantity of the longitudinal steel bars of the first layer of longitudinal columns, output the material list, and determine the top height l1 of the long steel bars of the first layer of high piles and the top height l2 of the long steel bars of the low piles;

[0008] Step 2: Determine the length, diameter, and quantity of the longitudinal steel bars of the longitudinal columns from the 2nd to the Nth layer layer by layer from bottom to top, and output the material list of the longitudinal steel bars; specifically, it includes:

[0009] Step 2.1: for the length of the raw steel bar, select n sub-raw steel bar lengths, the sub-raw steel bar lengths are cut from the raw steel bar and there is no waste in the raw steel bar;

[0010] Step 2.2: Determine the floor height, beam height and connectable area height range KLJ of the tth floor i , t=2,3,…,N;

[0011] Step 2.3: Select two lengths of l from the n raw material steel bar lengths. n1 and l n2 The selected sub-raw steel bar is connected to the top of the corresponding t-1 layer of longitudinal steel bar through the sleeve; the longitudinal steel bar height is iterated as l1=l1+l n1 and l2=l2+l n2 ;

[0012] Step 2.4: Determine whether the longitudinal reinforcement heights l1 and l2 are within the connectable area height range KLJ i At the same time, check whether the height difference l1-l2 of the raw steel bar meets the construction requirements; if not, return to step 2.3 to select the length of the sub-raw steel bar; if it meets the requirements, proceed to step 2.5;

[0013] Step 2.5: Repeat steps 2.2 to 2.4 until t=N, and output the blanking list of the 2nd to Nth layers of longitudinal steel bars;

[0014] Step 3: Determine the length of the longitudinal reinforcement connected to the roof beam and output the material list.

[0015] Furthermore, in step 1, the length of the longitudinal reinforcement of the first-layer longitudinal column is determined, specifically including:

[0016] Step 1.1: Obtain the cross-sectional dimensions of the longitudinal column and the longitudinal reinforcement data RF1, and calculate the length of the reinforcement connection area HK1;

[0017] Step 1.2: Determine the height range KLJ1 of the connectable area of ​​the first layer of longitudinal reinforcement;

[0018] Step 1.3: Divide the longitudinal reinforcement of the first layer into i1 The long steel bars of the high pile and l i2 The number of long steel bars of low piles is combined according to the diameter of the steel bars;

[0019] Step 1.4: Select the longitudinal steel bars with the same diameter, and the number of the merged high pile long steel bars and low pile long steel bars is n respectively. i1 and n i2, arrange the high-pile long steel bars and the low-pile long steel bars in the raw steel bars for cutting, obtain the scrap rate corresponding to the cutting scheme, and output the longitudinal steel bar length under the cutting scheme whose scrap rate meets the requirements;

[0020] Step 1.5: Repeat step 1.4 until the cutting plan for longitudinal steel bars of all diameters is completed and all longitudinal steel bar lengths are obtained.

[0021] Further, the longitudinal column is a rectangular column, and the cross-sectional dimensions of the longitudinal column include a long side length k1 and a short side length b1;

[0022] The longitudinal reinforcement of the longitudinal column includes angle reinforcement, long side reinforcement and short side reinforcement, wherein RF1 = {rfh1, rfb1, rfj1}, rfh1, rfb1, rfj1 are the reinforcement data of short side reinforcement, long side reinforcement and angle reinforcement respectively, wherein the angle reinforcement data rfj1 = {diameter rfjd1, quantity rfjn1}, the short side reinforcement data rfh1 = {diameter rfhd1, quantity rfhn1}, the long side reinforcement data rfb1 = {diameter rfbd1, quantity rfbn1};

[0023] HK1=35Max(rfjd1,rfbd1,rfhd1).

[0024] Further, KLJ1 = [hbu1 + mc1, hbd2 - mc1];

[0025] hbu1 is the top height of the first-layer beam, hbd2 is the bottom height of the second-layer beam, mc1 is the length of the non-connected area of ​​the first layer, where mc1=max(500mm,k1,Hn1 / 6), k1 is the long side length of the longitudinal column of the first layer, and Hn1 is the clear height of the first layer.

[0026] Furthermore, a genetic algorithm is used to optimize the steel bar cutting scheme in step 1.3, and the optimization variables are the type n3 and the number n of raw steel bars. k , the optimization goal is: i1 The root length is l 1,bottom The long steel bars of the high pile and n i2 The root length is l 2,bottom Find the raw steel bar cutting solution with the least scrap rate when cutting long steel bars with low piles;

[0027] Set the hyperparameters of the genetic algorithm, including population size P, mutation probability bp, and evolutionary generation Z;

[0028] The optimization objective function is:

[0029]

[0030] The optimization constraint function is:

[0031]

[0032] In the formula, n1 is the type of long steel bars in high piles and long steel bars in low piles, n k is the number of raw steel bars, n3 is the type of raw steel bars, is a positive integer; i,j is the selection coefficient, j means that the longitudinal reinforcement is arranged in the jth raw material reinforcement; i has two values, i1 means high pile long reinforcement, i2 means low pile long reinforcement; l i1 and l i2 L are the lengths of the high pile long steel bars and the low pile long steel bars respectively; k is the length of the kth raw steel bar.

[0033] Furthermore, during optimization, one of the diameters of the longitudinal reinforcement is selected and n i1 The root length is l 1,bottom The long steel bars of the high pile and n i2 The root length is l 2,bottom The long steel bars of the low pile are randomly arranged in n3 types, n k Generating P kinds of steel bar cutting schemes from the raw steel bars, wherein the cutting schemes satisfy the optimization constraint function;

[0034] The above P cutting schemes form an initial scheme library A0, and the initial fitness F0 of the population A0 is calculated according to the optimization objective function;

[0035] In the initial solution library A0, a solution with higher fitness is selected by roulette to participate in the reproduction of the next generation, and a population A1 of the next generation is generated after crossover and mutation operations, and the fitness F1 of the population A1 is calculated according to the optimization objective function;

[0036] Repeat the previous step until the maximum number of iterations is reached or the population fitness no longer changes significantly, and the steel bar cutting plan with the minimum scrap rate is obtained.

[0037] Furthermore, in step 2, the Markov state transfer optimization method is used to optimize the connection mode of the longitudinal reinforcement, specifically:

[0038] For the steel bar of the tth layer, when it is connected to the t+1th layer, its state transition is written as:

[0039] S t =(l 1t ,l 2t ) → S t+1 =(l 1t+1 ,l 2t+1 );

[0040] In the formula, l 1t and l 2tare the heights of the high pile long steel bars and low pile long steel bars of the tth layer, l 1t+1 and l 2t+1 are the heights of the high pile and low pile reinforcement bars of the t+1th layer, S t is the steel bar connection status of the tth layer, S t+1 is the steel bar connection status of the t+1th layer;

[0041] For the steel bar connection from the tth layer to the t+1th layer, the state transition action can be written as:

[0042] a t =(ΔL1,ΔL2),ΔL1,ΔL2∈{l1,...,l n1};

[0043] In the formula, {l1,…,l n1} is n different lengths of sub-raw steel bars;

[0044] Constraints on state transitions and state transition actions for:

[0045]

[0046] In the formula, l LB and l UB are the upper and lower limits of the connectable area of ​​the reinforcement in this layer, respectively. t The height difference of high and low pile reinforcement required by the structure;

[0047] The Markov state transfer optimization objective function of the optimal longitudinal reinforcement connection method is written as:

[0048] max E[∑R(S t ,a t ,S t+1 )];

[0049]

[0050] In the formula, E(·) is the mathematical expectation, R(S t ,a t ,S t+1 ) is the state transfer reward function.

[0051] Further, in step 3, the length of the longitudinal reinforcement connected to the roof beam is determined, specifically including:

[0052] Step 3.1: According to the iterated high pile long steel bar l1, low pile long steel bar l2 and roof height at t = N in step 2.5, calculate the length l of the high pile long steel bar on the top floor 1,top and low pile long steel bar 2,top , the number of long steel bars for high piles and long steel bars for low piles are merged according to the diameter of the longitudinal steel bars;

[0053] Step 3.2: Select the longitudinal steel bars with the same diameter, and record the number of long steel bars of the high pile obtained by merging as n. i1 The number of long steel bars in the low pile is recorded as n i2 ; and select n L1 The length of raw steel bar is L1 and n L2 The length of raw steel bar is L2, and the number is n i1 The length and number of high pile steel bars are n i2 The low pile long steel bars are arranged in the raw steel bars for cutting, the scrap rate corresponding to the cutting scheme is obtained, and the longitudinal steel bar length under the cutting scheme whose scrap rate meets the requirements is output;

[0054] Step 3.3: Repeat step 3.2 until the cutting plan for longitudinal steel bars of all diameters is completed and all longitudinal steel bar lengths are obtained.

[0055] Furthermore, a genetic algorithm is used to optimize the steel bar cutting scheme in step 3.2, and the optimization variables are the type n3 and the number n of raw steel bars. k , the optimization goal is: i1 The root length is l 1,top The long steel bars of the high pile and n i2 The root length is l 2,top Find the raw steel bar cutting solution with the least scrap rate when cutting long steel bars with low piles;

[0056] The hyperparameters of the genetic algorithm include population size P, mutation probability bp, and evolutionary generation Z;

[0057] The optimization objective function is:

[0058]

[0059] The optimization constraint function is:

[0060]

[0061] In the formula, n1 is the type of long steel bars in high piles and long steel bars in low piles, n k is the number of raw steel bars, n3 is the type of raw steel bars, is a positive integer; i,j is the selection coefficient, j means that the longitudinal reinforcement is arranged in the jth raw material reinforcement; i has two values, i1 means high pile long reinforcement, i2 means low pile long reinforcement; l i1 and l i2 L are the lengths of the high pile long steel bars and the low pile long steel bars respectively; k is the length of the kth raw steel bar.

[0062] Furthermore, when optimizing, n i1The root length is l 1,top The long steel bars of the high pile and n i2 The root length is l 2,top The long steel bars of the low pile are randomly arranged in n3 types, n k Generating P kinds of steel bar cutting schemes from the raw steel bars, wherein the cutting schemes satisfy the optimization constraint function;

[0063] The above P cutting schemes form an initial scheme library A0, and the initial fitness F0 of the population A0 is calculated according to the optimization objective function;

[0064] In the initial solution library A0, a solution with higher fitness is selected by roulette to participate in the reproduction of the next generation, and a population A1 of the next generation is generated after crossover and mutation operations, and the fitness F1 of the population A1 is calculated according to the optimization objective function;

[0065] Repeat the previous step until the maximum number of iterations is reached or the population fitness no longer changes significantly, and the steel bar cutting plan with the minimum scrap rate is obtained.

[0066] Due to the adoption of the above technical solution, the present invention has the following advantages and positive effects compared with the prior art: the present invention calculates the lengths of the longitudinal steel bars of the longitudinal columns of the bottom layer and the top layer separately, and calculates the longitudinal steel bars of the longitudinal columns of the middle layer layer by layer from bottom to top using the same calculation method, so that the splicing position of the longitudinal steel bars is within the height range of the connectable area KLJ i and make the height difference between the long steel bars of high piles and the long steel bars of low piles meet the requirements. Moreover, when the Markov state transition optimization method is used to optimize the connection method of the middle layer of longitudinal steel bars, the optimal longitudinal steel bar splicing scheme and the middle layer longitudinal steel bar cutting list can be obtained. In addition, for the bottom layer and top layer of steel bars, the present invention also uses a genetic algorithm to optimize the cutting scheme of the longitudinal steel bars, and can obtain a raw steel bar cutting scheme with the lowest scrap rate. The method for calculating the cutting length of the column longitudinal steel bars provided by the present invention comprehensively considers the optimization of the cutting and pattern making of the bottom layer of longitudinal steel bars, the middle layer of steel bars and the top layer of longitudinal steel bars, improves the efficiency of the longitudinal steel bar pattern making, and saves materials while meeting compliance and constructability. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 It is a flow chart of a method for calculating the cutting length of longitudinal reinforcement bars of a column in one embodiment of the present invention;

[0068] Figure 2 It is a schematic diagram of the bottom beam, longitudinal column and longitudinal reinforcement in one embodiment of the present invention;

[0069] Figure 3 is a schematic diagram of an intermediate layer beam, a longitudinal column and longitudinal reinforcement in one embodiment of the present invention;

[0070] Figure 4 Schematic diagram of roof beams, longitudinal columns and longitudinal reinforcements in one embodiment of the present invention;

[0071] Figure 5 is a cross-sectional view of a longitudinal column of floor t in one embodiment of the present invention;

[0072] Figure 6 is a cross-sectional view of a longitudinal column of floor t-1 in one embodiment of the present invention;

[0073] Figure 7 Schematic diagram of a steel bar cutting scheme in one embodiment of the present invention.

[0074] The numbers in the figure are as follows:

[0075] 1-longitudinal column; 2-beam; 3-longitudinal reinforcement; 4-connection node; 5-angle reinforcement; 6-long side reinforcement; 7-short side reinforcement. DETAILED DESCRIPTION

[0076] The following is a further detailed description of a method for calculating the length of a column longitudinal reinforcement provided by the present invention in conjunction with the accompanying drawings and specific embodiments. The advantages and features of the present invention will become clearer with the following description. It should be noted that the accompanying drawings are in a very simplified form and are not in precise proportions, and are only used to conveniently and clearly assist in explaining the purpose of the embodiments of the present invention.

[0077] like Figure 1 As shown, a method for calculating the cutting length of longitudinal steel bars of a column provided in this embodiment is used to determine the cutting length of longitudinal steel bars of a longitudinal column of a building. The building includes N floors, such as Figure 2 As shown, the bottom of the longitudinal reinforcement 3 of the first-layer longitudinal column is fixed in the bottom beam 2, and the top is connected to the bottom of the longitudinal reinforcement 3 of the second layer through a sleeve, and the connection node 4 is located within the height range KLJ1 of the connectable area of ​​the first layer; combined with Figure 2 and Figure 3 As shown, the bottom of the longitudinal reinforcement 3 of the longitudinal column 1 of the tth layer is connected to the top of the longitudinal reinforcement of the longitudinal column of the t-1th layer through a sleeve, that is, the top connection node 4 of the longitudinal reinforcement of the longitudinal column of the tth layer is located in the connectable area height range KLJ above the beam (or floor slab) of the tth layer. t KLJ with the bottom below the t-th floor beam (or floor slab) t-1 In the above, i = 2, 3, ..., N; in addition, the longitudinal reinforcement connected to the roof beam above the Nth layer longitudinal column is also involved, such as Figure 4As shown, the top of the longitudinal reinforcement is anchored in the roof beam. The first layer can be called the bottom layer, the second to N layers can be called the middle layer, and the longitudinal reinforcement connecting the Nth layer of longitudinal columns to the roof can be called the top layer. In this embodiment, the length of the longitudinal reinforcement of the longitudinal columns of the bottom layer and the top layer is calculated separately, and the longitudinal reinforcement of the longitudinal columns of the middle layer is calculated layer by layer using the same calculation method. The method for calculating the cutting length of the column longitudinal reinforcement provided in this embodiment includes the following steps:

[0078] Step 1: determine the length, diameter, and quantity of the first layer of column longitudinal steel bars, and output the material list to determine the top height l1 of the first layer of high pile long steel bars and the top height l2 of the low pile long steel bars;

[0079] Step 2: Determine the length of the longitudinal reinforcement of the longitudinal column of the tth layer, t = 2, 3, ..., N, and summarize and output the material list; specifically include:

[0080] Step 2.1: For raw steel bars with lengths of L1 and L2, select n sub-raw steel bar lengths, which are cut from raw steel bars with lengths of L1 and L2 and do not contain any waste, such as L1 = 9m, L2 = 12m; the selected sub-raw steel bar length can be completely cut from raw steel bars with lengths of L1 = 9m or L2 = 12m without any waste, and the sub-raw steel bar length can be any one of [2m, 3m, 4.5m, 6m];

[0081] Step 2.2: Determine the floor height, beam height and connectable area height range KLJ of the tth floor t ; For example, KLJ t =[hbu t +mc t ,hbd t -mc t ], hbu t is the top height of the beam on the i-th layer, hbd i+1 is the bottom height of the i+1th layer beam, mc i is the length of the non-connected area of ​​the i-th layer, where mc1 = max(500mm, k i ,Hn i / 6), k i is the long side length of the i-th layer longitudinal column, Hn i is the net height of the tth floor; for example, according to the net height of the second floor, the height range of the connectable area of ​​the second floor is calculated to be KLJ2 = [6.6m, 9.2m];

[0082] Step 2.3: Select two lengths of l from the n raw material steel bar lengths. n1 and l n2The selected sub-raw steel bar is connected to the top of the longitudinal steel bar of the corresponding t-1th layer longitudinal column through the sleeve; at this time, the longitudinal steel bar height is iterated to l1=l1+l n1 and l2=l2+l n2 ; For example, when the height of the first layer of longitudinal reinforcement is l1=1.3m and l2=2.3m, select two steel bar lengths from the four sub-raw material reinforcement lengths [2m, 3m, 4.5m, 6m], and connect the selected sub-raw material reinforcement to the first layer of longitudinal reinforcement through sleeves; at this time, the top height of the second layer of longitudinal reinforcement is iteratively l1=l1+l n1 and l2=l2+l n2 ;

[0083] Step 2.4: Determine whether the longitudinal reinforcement heights l1 and l2 are within the connectable area height range KLJ t At the same time, check whether the height difference l1-l2 of the raw steel bars meets the construction requirements (i.e., greater than the length of the connection area); if not, return to step 2.3 to select the length of the sub-raw steel bars; if it meets the requirements, proceed to step 2.5; for example, when l1=1.3m and l2=2.3m, select a steel bar with a length of 4.5m, only the high pile long steel bar is within the interval of KLJ2, and the low pile long steel bar is not within the interval of KLJ2, and it is necessary to return to reselect; select a steel bar with a length of 6m, both the high pile long steel bar and the low pile long steel bar are within the interval of KLJ2, and you can proceed to step 2.5;

[0084] Step 2.5: Repeat steps 2.2 to 2.4 until t=N, and output the blanking list of the 2nd to Nth layers of longitudinal steel bars;

[0085] Step 3: Determine the length of the longitudinal reinforcement connected to the roof beam and output the material list.

[0086] In a specific embodiment, after step 3, the method further includes:

[0087] Step 4: Increase or decrease the lengths l1 and l2 of the longitudinal reinforcement of the first-layer longitudinal column in step 1 at intervals of 0.1m within the height range KLJ1 of the connectable area, repeat steps 1 to 3 until all possible longitudinal reinforcement lengths l1 and l2 within the height range KLJ1 of the connectable area are traversed, output different reinforcement connection schemes, and compare all reinforcement connection schemes, select the reinforcement cutting scheme with the least waste, and output the optimal material cutting list. It should be noted that there may be multiple reinforcement cutting schemes with the least waste, and the reinforcement cutting scheme results obtained in steps 1 to 4 are not unique. Construction personnel can select a suitable re-sampling scheme suitable for the actual situation on site from the several schemes calculated by the program according to the actual situation on site, and obtain the reinforcement cutting list corresponding to the re-sampling scheme.

[0088] The following is an introduction using a rectangular column as an example. Figures 1 to 6 As shown, the longitudinal column is a rectangular column, which intersects with the beam. The cross section of the rectangular column of the tth layer includes the long side length k t and the short side length b t , the longitudinal reinforcement 3 of the longitudinal column 1 includes angle reinforcement 5, long side reinforcement 6 and short side reinforcement 7, among which the angle reinforcement data, long side reinforcement data and short side reinforcement data of the t floor are respectively recorded as rfj t 、rfb t 、rfh t , where rfj t ={diameter rfjd t , quantity rfjn t}, rfb t ={diameter rfbd t , quantity rfbn t}, rfh t ={diameter rfhd t , quantity rfhn t}.like Figure 5 As shown, the number of corner bars is 4, the number of long side bars is 2, and the number of short side bars is 2. Figure 6 As shown, the number of corner bars is 4, the number of long side bars is 4, and the number of short side bars is 8. The calculation method is further described below by taking a rectangular column as an example.

[0089] In step 1, the length of the longitudinal reinforcement of the first-layer longitudinal column needs to satisfy that the heights of l1 and l2 are within the height range KLJ1 of the first-layer connectable area and the height difference meets the requirements, and also meets the requirements of the bottom anchorage length, and minimizes the waste of reinforcement raw materials. The length of the longitudinal reinforcement of the first-layer longitudinal column can be determined in a traditional way. As a preferred way, this embodiment provides an optimization scheme for the first-layer longitudinal reinforcement, including the following steps:

[0090] Step 1.1: Obtain the cross-sectional dimensions of the longitudinal column and the longitudinal steel bar reinforcement data RF1, and calculate the steel bar connection area length HK1; wherein the cross-sectional dimensions of the longitudinal column include the long side length k1 and the short side length b1, wherein RF1 = {rfh1, rfb1, rfj1}, rfh1, rfb1, rfj1 are the reinforcement data of the short side bars, long side bars, and angle bars, respectively, wherein the angle bar data rfj1 = {diameter rfjd1, quantity rfjn1}, the short side bar data rfh1 = {diameter rfhd1, quantity rfhn1}, the long side bar data rfb1 = {diameter rfbd1, quantity rfbn1}, HK1 = 35Max(rfjd1, rfbd1, rfhd1). For example, according to statistics, there are 50 longitudinal columns on the first floor of a building. According to the drawings, there are rfjn1 = 200 corner bars with a diameter of rfjd1 = 28 mm, rfhn1 = 200 short side bars with a diameter of rfhd1 = 25 mm, and rfbn1 = 200 long side bars with a diameter of rfbd1 = 25 mm. The length of the steel bar connection area is HK1 = 35max (rfjd1, rfbd1, rfhd1) = 0.98 m. For the convenience of construction, it can be rounded up to 1.0 m. The length of the steel bar connection area is recorded as HK i , is the height at which the tops of the low pile long steel bars and the high pile long steel bars should be staggered.

[0091] Step 1.2: Figure 2 As shown, determine the height range KLJ1 of the connectable area of ​​the first layer of longitudinal reinforcement. Among them, KLJ1=[hbu1+mc1,hbd2-mc1]. hbu1 is the top height of the first layer of beams hbu1, hbd2 is the bottom height of the second layer of beams, mc1 is the length of the non-connected area of ​​the first layer, among which mc1=max(500mm,k1,Hn1 / 6), k1 is the long side length of the first layer of longitudinal columns, and Hn1 is the net height of the first layer. For example, the top height of the first layer of beams hbu1=0m, the low height of the second layer of beams hbd2=5.4m, Hn1=hbd2–hbu1=5.4m; mc1=max(500mm,600mm,5.4m / 6)=0.9m, KLJ1=[hbu0+mc1,hbd1-mc1]=[0.9m,4.5m].

[0092] Step 1.3: If Figure 7 As shown, the longitudinal reinforcement of the first layer is divided into i1 The long steel bars of the high pile and l i2 The number of high pile long steel bars and low pile long steel bars are merged according to the steel bar diameter; among them, i1 represents the high pile long steel bar and i2 represents the low pile long steel bar.

[0093] Step 1.4: Select the longitudinal steel bars with the same diameter, and the number of the merged high pile long steel bars and low pile long steel bars is n respectively. i1 and n i2 , arrange the high pile long steel bars and low pile long steel bars in the raw steel bars for cutting, obtain the scrap rate corresponding to the cutting scheme, and output the longitudinal steel bar length under the cutting scheme with the scrap rate meeting the requirements. Taking the longitudinal steel bar with a diameter of rfjd1 as an example, select n L1 The root length is L1, n L2 The length of raw steel bar is L2, generally L1 = 9m, L2 = 12m, the number is n i1 、n i2 The high-pile long steel bars and low-pile long steel bars are arranged in the raw steel bars for cutting, and the scrap rate corresponding to the cutting plan is obtained.

[0094] Step 1.5: Repeat step 1.4 until the cutting schemes of all diameters of longitudinal steel bars are completed and all longitudinal steel bar lengths are obtained. For longitudinal steel bars with diameters of rfjd1, rfhd1 and rfbd1, all corresponding cutting schemes and scrap rates corresponding to the cutting schemes are obtained.

[0095] Furthermore, a genetic algorithm is used to optimize the steel bar cutting scheme in step 1.4, and the optimization variables are the type n3 and the number n of raw steel bars. k , the optimization goal is: i1 The root length is l 1,bottom The long steel bars of the high pile and n i2 The root length is l 2,bottom When the length of the steel bar is low, find the cutting scheme with the least scrap rate. The type of raw steel bar n3, for example, all the raw steel bars with a length of L1 are used for cutting, all the raw steel bars with a length of L2 are used for cutting, or some of the raw steel bars with a length of L1 are used for cutting, and some of the raw steel bars with a length of L2 are used for cutting. The number of each type of steel bar in the case of the type of raw steel bar n3 is recorded as n k The hyperparameters of the genetic algorithm that need to be set include the population size P, the mutation probability bp, and the evolution generation Z. During optimization, it is necessary to ensure that the steel bar connection area is within the range KLJ1, and the top height difference of the high pile long steel bar and the low pile long steel bar is equal to the connection length HK1. For example, call the genetic algorithm to optimize the steel bar with a diameter of 28mm in step 1.3. The optimization variables are the type n3 and the number n of raw steel bars. k , the optimization goal is, in n i1 = 100 roots with a length of l 1,bottom = 2.5m high pile length steel bar and n i2 = 100 roots with a length of l 2,bottom=3.5m low pile length steel bar to find the raw steel bar cutting plan with the lowest scrap rate. Set the hyper parameters of the genetic algorithm, including population size P = 100, mutation probability bp = 0.01, and evolution generation Z = 100. During optimization, it is necessary to ensure that the steel bar connection area is within the range KLJ1 = [0.9m, 4.5m], and the difference in the length of the high and low piles is equal to the connection length HK1 = 1.0m.

[0096] The optimization objective function is shown in formula (1):

[0097]

[0098] The optimization constraint function is shown in formula (2):

[0099]

[0100] In the formula, n1 is the type of long steel bars in high piles and long steel bars in low piles, n k is the number of raw steel bars, n3 is the type of raw steel bars, both are positive integers y i,j is the selection coefficient, j means that the current steel bar is arranged in the jth raw steel bar; i has two values, i1 means that the steel bar is a high pile long steel bar, and i2 means that the steel bar is a low pile long steel bar; l i1 and l i2 L are the lengths of the high pile long steel bars and the low pile long steel bars respectively; k is the length of the kth raw steel bar. In actual projects, the length of the raw steel bar is usually 9m or 12m.

[0101] The optimization process is as follows:

[0102] n i1 The root length is l 1,bottom The long steel bars of the high pile and n i2 The root length is l 2,bottom The long steel bars of the low pile are randomly arranged in n3 types, n k P kinds of steel bar cutting schemes are formed from the raw steel bars, and the cutting schemes need to satisfy the constraints shown in formula (2);

[0103] The above P cutting schemes form the initial scheme library A0, and the initial fitness F0 of the population A0 is calculated according to formula (1);

[0104] In the initial solution library A0, a solution with higher fitness is selected by roulette to participate in the reproduction of the next generation. After crossover and mutation operations, the next generation population A1 is generated. The fitness F1 of population A1 is calculated according to formula (1);

[0105] Repeat the above steps until the maximum number of iterations is reached or the population fitness no longer changes significantly, and the steel bar cutting scheme with the minimum scrap rate is obtained. At this time, the number of raw steel bars is: The first raw steel bar n k,1 Root, the second raw material steel bar n k,2 Root, until the n3rd raw material steel bar n k,n3 The corresponding optimal steel bar cutting solution can be recorded as A max .

[0106] Step 1.5: According to the different steel bar diameters rfjd1, rfbd1, rfhd1, the length l is given respectively 1,bottom The bottom high pile has a long steel bar and a length of l 2,bottom Output the steel bar cutting plan according to the length of the low pile long steel bar. The top elevations of the high pile long steel bar and the low pile long steel bar are recorded as l1 and l2 respectively.

[0107] In a specific embodiment, in step 2, the Markov state transition optimization method is used to determine the optimal steel bar connection method. For the longitudinal steel bars of the tth layer, when they are connected to the t+1th layer, their state transition can be written as:

[0108] S t =(l 1t ,l 2t ) → S t+1 =(l 1t+1 ,l 2t+1 ) (3)

[0109] In the formula, l 1t and l 2t are the height of the long steel bars of the high pile and the low pile in the tth layer, respectively, 1t+1 and l 2t+1 are the top heights of the high pile long steel bars and low pile long steel bars of the t+1th layer, S t is the steel bar connection status of the tth layer, S t+1 It is the steel bar connection status of the t+1th layer.

[0110] For the steel bar connection from the tth layer to the t+1th layer, the state transition action can be written as:

[0111] a t =(ΔL1,ΔL2),ΔL1,ΔL2∈{l1,...,l n} (4)

[0112] In the formula, {l1,…,l n} are n different sub-raw material steel bar lengths.

[0113] For the state transfer equations of formula (3) and formula (4), the constraints are for:

[0114]

[0115] In the formula, l LB and l UB are the upper and lower limits of the connectable area of ​​t+1 steel bars, d t To construct the required height difference between the high-length steel bars and the low-length steel bars, refer to step 1.1 and calculate it based on the steel bar diameter data of the current layer.

[0116] According to equations (3) to (5), the Markov state transfer optimization objective function of the optimal reinforcement connection method can be written as:

[0117] max E[∑R(S t ,a t ,S t+1 )] (6)

[0118]

[0119] In the formula, E(·) is the mathematical expectation, R(S t ,a t ,S t+1 ) is the state transfer reward function.

[0120] This embodiment can complete the connection optimization of the middle layer longitudinal steel bars, and the optimization steps of each step in the middle are omitted here. As an example, in a certain construction, after the optimization is completed, the top heights of the low pile long steel bars and high pile long steel bars of the Nth layer are 153.8m and 154.8m respectively.

[0121] In a specific embodiment, in step 3, determining the length of the column longitudinal reinforcement connecting the top of the Nth layer of longitudinal reinforcement and the roof specifically includes the following steps:

[0122] Step 3.1: For the top steel bars with heights of l1 and l2 after step 2.5, their connection area and connection height difference meet the requirements, and only the problem of breaking the top steel bars needs to be considered. Calculate the length l of the top high pile steel bar according to the top elevation of the floor 1,top and the length of the long steel bar of the low pile l 2,top , the diameter of the top angle reinforcement, short side reinforcement and long side reinforcement is recorded as rfjd end ,rfbd end ,rfhd end The number of high pile long steel bars and low pile long steel bars are merged according to the steel bar diameter. For steel bars with the same diameter, the number of high pile long steel bars and low pile long steel bars obtained by merging are n and n respectively. i1 and n i2 .

[0123] Step 3.2:

[0124] Step 3.2: Select the longitudinal steel bars with the same diameter, and record the number of long steel bars of the high pile obtained by merging as n. i1 The number of long steel bars in the low pile is recorded as n i2 ; and select n L1 The length of raw steel bar is L1 and n L2 The length of raw steel bar is L2, and the number is n i1 The length and number of high pile steel bars are n i2 The low pile long steel bars are arranged in the raw steel bars for cutting, and the scrap rate corresponding to the cutting scheme is obtained, and the longitudinal steel bar length under the cutting scheme with the scrap rate meeting the requirements is output. For example, for a diameter of rfjd end For longitudinal reinforcement, select n L1 The length of raw steel bar is L1 and n L2 The length of raw steel bar is L2 (generally L1 = 9m, L2 = 12m), and the number is n i1 The length and number of high pile steel bars are n i2 The low-pile long steel bars are arranged and cut among the raw steel bars to obtain the scrap rate corresponding to the cutting plan, and determine whether the scrap rate meets the requirements. If it meets the requirements, proceed to the next step. If it does not meet the requirements, rearrange it.

[0125] Step 3.3: Repeat step 3.2 until the cutting plan of longitudinal reinforcements of all diameters is completed and all longitudinal reinforcement lengths are obtained. end 、rfhd end and rfbd end For the longitudinal steel bars, all the corresponding cutting plans and the scrap rates corresponding to the cutting plans are obtained.

[0126] Furthermore, a genetic algorithm is used to optimize the problem described in step 3.2, and the optimization variables are the type n3 and the number n of raw steel bars. k , the optimization goal is: i1 The root length is l 1,top The long steel bars of the high pile and n i2 The root length is l 2,top When the length of the steel bar is low, the cutting scheme of the raw steel bar with the least scrap rate is found. The hyper parameters of the genetic algorithm that need to be set include the population size P, the mutation probability bp, and the evolutionary generation Z. The optimization objective function is shown in formula (8):

[0127]

[0128] The optimization constraint function is shown in formula (9):

[0129]

[0130] In the formula, n1 is the type of long steel bars in high piles and long steel bars in low piles, nk is the number of raw steel bars, n3 is the type of raw steel bars, is a positive integer; i,j is the selection coefficient, j means that the current steel bar is arranged in the jth raw steel bar; i has two values, i1 means high pile long steel bar, i2 means low pile long steel bar; l i1 and l i2 L are the lengths of the high pile long steel bars and the low pile long steel bars respectively; k is the length of the kth raw steel bar. In actual projects, the length of the raw steel bar is usually 9m or 12m.

[0131] The optimization process is as follows:

[0132] n i1 The root length is l 1,top The long steel bars of the high pile and n i2 The root length is l 2,top The long steel bars of the low pile are randomly arranged in n3 types, n k P kinds of steel bar cutting schemes are formed from the raw steel bars, and the cutting schemes need to satisfy the constraints shown in formula (2);

[0133] The above P cutting schemes form the initial scheme library A0, and the initial fitness F0 of the population A0 is calculated according to formula (1);

[0134] In the initial solution library A0, a solution with higher fitness is selected by roulette to participate in the reproduction of the next generation. After crossover and mutation operations, the next generation population A1 is generated. The fitness F1 of population A1 is calculated according to formula (1);

[0135] Repeat the above steps until the maximum number of iterations is reached or the population fitness no longer changes significantly, and the steel bar cutting scheme with the minimum scrap rate is obtained. At this time, the number of raw steel bars is the first raw steel bar n k,1 Root, the second raw material steel bar n k,2 Root, until the n3rd raw material steel bar n k,n3 The corresponding optimal steel bar cutting solution is recorded as A max .

[0136] Step 3.4: According to different steel bar diameters rfjd end ,rfbd end ,rfhd end Give the length l respectively 1,top The long steel bars of the high pile and the length are l 2,top The raw steel bar cutting plan for low pile long steel bars can be output as steel bar cutting list.

[0137] The technical features of the above-described embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0138] The above-mentioned embodiments only express several implementation modes of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the invention. It should be pointed out that, for a person of ordinary skill in the art, several modifications and improvements can be made without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the attached claims.

Claims

1. A method for calculating the length of longitudinal steel bars in a column, characterized in that: The building comprises N floors, a beam intersecting with a longitudinal column is arranged at the bottom of each floor, a roof beam is arranged at the roof of the building, and the longitudinal reinforcement of the longitudinal column comprises a high pile long reinforcement and a low pile long reinforcement; the calculation method comprises the following steps: Step 1: determine the length, diameter, and quantity of the longitudinal steel bars of the first layer of longitudinal columns, output the material list, and determine the top height l1 of the long steel bars of the first layer of high piles and the top height l2 of the long steel bars of the low piles; Step 2: Determine the length, diameter, and quantity of the longitudinal steel bars of the longitudinal columns from the 2nd to the Nth layer layer by layer from bottom to top, and output the material list of the longitudinal steel bars; specifically, it includes: Step 2.1: for the length of the raw steel bar, select n sub-raw steel bar lengths, the sub-raw steel bar lengths are cut from the raw steel bar and there is no waste in the raw steel bar; Step 2.2: Determine the floor height, beam height and connectable area height range KLJ of the tth floor i , t=2,3,…,N; Step 2.3: Select two lengths of l from the n raw material steel bar lengths. n1 and l n2 The selected sub-raw steel bar is connected to the top of the corresponding t-1 layer of longitudinal steel bar through the sleeve; the longitudinal steel bar height is iterated to l1=l1+l n1 and l2=l2+l n2 ; Step 2.4: Determine whether the longitudinal reinforcement heights l1 and l2 are within the connectable area height range KLJ i At the same time, check whether the height difference l1-l2 of the raw steel bar meets the construction requirements; if not, return to step 2.3 to select the length of the sub-raw steel bar; if it meets the requirements, proceed to step 2.5; Step 2.5: Repeat steps 2.2 to 2.4 until t=N, and output the blanking list of the 2nd to Nth layers of longitudinal steel bars; Step 3: Determine the length of the longitudinal reinforcement connected to the roof beam and output the material list.

2. The method for calculating the length of the longitudinal reinforcement of a column according to claim 1, characterized in that: In step 1, the length of the longitudinal reinforcement of the first-layer longitudinal column is determined, including: Step 1.1: Obtain the cross-sectional dimensions of the longitudinal column and the longitudinal reinforcement data RF1, and calculate the length of the reinforcement connection area HK1; Step 1.2: Determine the height range KLJ1 of the connectable area of ​​the first layer of longitudinal reinforcement; Step 1.3: Divide the longitudinal reinforcement of the first layer into i1 The long steel bars of the high pile and l i2 The number of long steel bars of low piles is combined according to the diameter of the steel bars; Step 1.4: Select the longitudinal steel bars with the same diameter, and the number of the merged high pile long steel bars and low pile long steel bars is n respectively. i1 and n i2 , arrange the high-pile long steel bars and the low-pile long steel bars in the raw steel bars for cutting, obtain the scrap rate corresponding to the cutting scheme, and output the longitudinal steel bar length under the cutting scheme whose scrap rate meets the requirements; Step 1.5: Repeat step 1.4 until the cutting plan for longitudinal steel bars of all diameters is completed and all longitudinal steel bar lengths are obtained.

3. The method for calculating the length of the column longitudinal reinforcement cutting according to claim 2, characterized in that: The longitudinal column is a rectangular column, and the cross-sectional dimensions of the longitudinal column include a long side length k1 and a short side length b1; The longitudinal reinforcement of the longitudinal column includes angle reinforcement, long side reinforcement and short side reinforcement, wherein RF1 = {rfh1, rfb1, rfj1}, rfh1, rfb1, rfj1 are the reinforcement data of short side reinforcement, long side reinforcement and angle reinforcement respectively, wherein the angle reinforcement data rfj1 = {diameter rfjd1, quantity rfjn1}, the short side reinforcement data rfh1 = {diameter rfhd1, quantity rfhn1}, the long side reinforcement data rfb1 = {diameter rfbd1, quantity rfbn1}; HK1=35Max(rfjd1,rfbd1,rfhd1).

4. The method for calculating the length of the longitudinal reinforcement of a column as claimed in claim 3, characterized in that: KLJ1=[hbu1+mc1,hbd2-mc1]; hbu1 is the top height of the first-layer beam, hbd2 is the bottom height of the second-layer beam, mc1 is the length of the non-connected area of ​​the first layer, where mc1=max(500mm,k1,Hn1 / 6), k1 is the long side length of the longitudinal column of the first layer, and Hn1 is the clear height of the first layer.

5. The method for calculating the length of the column longitudinal reinforcement cutting according to claim 3, characterized in that: Genetic algorithm is used to optimize the steel bar cutting scheme in step 1.

3. The optimization variables are the type n3 and the number n of raw steel bars. k , the optimization goal is: i1 The root length is l 1,bottom The long steel bars of the high pile and n i2 The root length is l 2,bottom Find the raw steel bar cutting solution with the least scrap rate when cutting long steel bars with low piles; Set the hyperparameters of the genetic algorithm, including population size P, mutation probability bp, and evolutionary generation Z; The optimization objective function is: The optimization constraint function is: In the formula, n1 is the type of long steel bars in high piles and long steel bars in low piles, n k is the number of raw steel bars, n3 is the type of raw steel bars, is a positive integer; i,j is the selection coefficient, j means that the longitudinal reinforcement is arranged in the jth raw material reinforcement; i has two values, i1 means high pile long reinforcement, i2 means low pile long reinforcement; l i1 and l i2 L are the lengths of the high pile long steel bars and the low pile long steel bars respectively; k is the length of the kth raw steel bar.

6. The method for calculating the length of the longitudinal reinforcement of a column according to claim 5, characterized in that: When optimizing, select one of the diameters of the longitudinal reinforcement and set n i1 The root length is l 1,bottom The long steel bars of the high pile and n i2 The root length is l 2,bottom The long steel bars of the low pile are randomly arranged in n3 types, n k Generating P kinds of steel bar cutting schemes from the raw steel bars, wherein the cutting schemes satisfy the optimization constraint function; The above P cutting schemes form an initial scheme library A0, and the initial fitness F0 of the population A0 is calculated according to the optimization objective function; In the initial solution library A0, a solution with higher fitness is selected by roulette to participate in the reproduction of the next generation, and a population A1 of the next generation is generated after crossover and mutation operations, and the fitness F1 of the population A1 is calculated according to the optimization objective function; Repeat the previous step until the maximum number of iterations is reached or the population fitness no longer changes significantly, and the steel bar cutting plan with the minimum scrap rate is obtained.

7. The method for calculating the length of the column longitudinal reinforcement cutting according to claim 1, characterized in that: In step 2, the Markov state transfer optimization method is used to optimize the connection mode of the longitudinal reinforcement, specifically: For the steel bar of the tth layer, when it is connected to the t+1th layer, its state transition is written as: S t =(l 1t ,L 2t )→S t+1 =(l 1t+1 ,L 2t+1 ); In the formula, l 1t and l 2t are the heights of the high pile long steel bars and low pile long steel bars of the tth layer, l 1t+1 and l 2t+1 are the heights of the high pile and low pile reinforcement bars of the t+1th layer, S t is the steel bar connection status of the tth layer, S t+1 is the steel bar connection status of the t+1th layer; For the steel bar connection from the tth layer to the t+1th layer, the state transition action can be written as: a t =(ΔL1,ΔL2),ΔL1,ΔL2∈{l1,...,l n1 }; In the formula, {l1,…,l n1 } is n different lengths of sub-raw steel bars; Constraints on state transitions and state transition actions for: In the formula, l LB and l UB are the upper and lower limits of the connectable area of ​​the reinforcement in this layer, respectively. t The height difference of high and low pile reinforcement required by the structure; The Markov state transfer optimization objective function of the optimal longitudinal reinforcement connection method is written as: max E[∑R(S t ,a t ,S t+1 )]; In the formula, E(·) is the mathematical expectation, R(S t ,a t ,S t+1 ) is the state transfer reward function.

8. The method for calculating the length of the longitudinal reinforcement of a column according to claim 1, characterized in that: In step 3, the length of the longitudinal reinforcement connected to the roof beam is determined, including: Step 3.1: According to the iterated high pile long steel bar l1, low pile long steel bar l2 and roof height at t = N in step 2.5, calculate the length l of the high pile long steel bar on the top floor 1,top and low pile long steel bar 2,top , the number of long steel bars for high piles and long steel bars for low piles are merged according to the diameter of the longitudinal steel bars; Step 3.2: Select the longitudinal steel bars with the same diameter, and record the number of long steel bars of the high pile as n. i1 The number of long steel bars in the low pile is recorded as n i2 ; and select n L1 Raw steel bar with length L1 and n L2 The length of raw steel bar is L2, and the number is n i1 The length and number of high pile steel bars is n i2 The low pile long steel bars are arranged in the raw steel bars for cutting, the scrap rate corresponding to the cutting scheme is obtained, and the longitudinal steel bar length under the cutting scheme whose scrap rate meets the requirements is output; Step 3.3: Repeat step 3.2 until the cutting plan for longitudinal steel bars of all diameters is completed and all longitudinal steel bar lengths are obtained.

9. The method for calculating the length of the column longitudinal reinforcement cutting according to claim 8, characterized in that: Genetic algorithm is used to optimize the steel bar cutting scheme in step 3.

2. The optimization variables are the type n3 and the number n of raw steel bars. k , the optimization goal is: i1 The root length is l 1,top The long steel bars of the high pile and n i2 The root length is l 2,top Find the raw steel bar cutting solution with the least scrap rate when cutting long steel bars with low piles; The hyperparameters of the genetic algorithm include population size P, mutation probability bp, and evolutionary generation Z; The optimization objective function is: The optimization constraint function is: In the formula, n1 is the type of long steel bars in high piles and long steel bars in low piles, n k is the number of raw steel bars, n3 is the type of raw steel bars, is a positive integer; i,j is the selection coefficient, j means that the longitudinal reinforcement is arranged in the jth raw material reinforcement; i has two values, i1 means high pile long reinforcement, i2 means low pile long reinforcement; l i1 and l i2 L are the lengths of the high pile long steel bars and the low pile long steel bars respectively; k is the length of the kth raw steel bar.

10. The method for calculating the length of the column longitudinal reinforcement cutting according to claim 9, characterized in that: When optimizing, n i1 The root length is l 1,top The long steel bars of the high pile and n i2 The root length is l 2,top The long steel bars of the low pile are randomly arranged in n3 types, n k Generating P kinds of steel bar cutting schemes from the raw steel bars, wherein the cutting schemes satisfy the optimization constraint function; The above P cutting schemes form an initial scheme library A0, and the initial fitness F0 of the population A0 is calculated according to the optimization objective function; In the initial solution library A0, a solution with higher fitness is selected by roulette to participate in the reproduction of the next generation, and a population A1 of the next generation is generated after crossover and mutation operations, and the fitness F1 of the population A1 is calculated according to the optimization objective function; Repeat the previous step until the maximum number of iterations is reached or the population fitness no longer changes significantly, and the steel bar cutting plan with the minimum scrap rate is obtained.

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