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Journal of Asian Architecture and Building Engineering
, Volume 3 (1): 7 – May 1, 2004

/lp/taylor-francis/algorithms-for-reducing-the-waste-rate-of-reinforcement-bars-OlrK8QL6s3

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- Publisher
- Taylor & Francis
- Copyright
- © 2018 Architectural Institute of Japan
- ISSN
- 1347-2852
- eISSN
- 1346-7581
- DOI
- 10.3130/jaabe.3.17
- Publisher site
- See Article on Publisher Site

Loss of rebars can be minimized with minimum use of discrete bars in market length. In order to achieve this goal, the accurate and detailed information of rebars is extracted, followed by both rapid and efficient bar combination. No paper has dealt directly with the reduction of rebar waste rates, although many researches have proposed indirect approaches to enhance productivity, constructability, safety and quality in the process of concrete reinforcement work. This paper, therefore, was prepared with the aim of developing algorithms to supply rebars required to minimize material waste during cutting and bending of discrete bars in rebar shops. At the same time, this study presented an automatic rebar detailing concept, a logical process of rebar combination with pertinent algorithms and binary search algorithm for bar data to implement the proposed topic. The effectiveness of the suggested algorithms was validated by case studies. Keywords: rebar work; optimization; algorithm; waste rate; combination Introduction The manual bar combination of either market length Countries with highly capital-intensive construction or special length to minimize the loss of materials is not use the computerized numerically controlled (CNC) an easy task since rebars of different diameters, lengths, machine. The machine automatically produces shaped and locations are found in various locations in drawings. rebars of up to 16 mm in diameter supplied in coils which In particular, in the rebar shop, the simultaneous is described as machine type A by Navon, Rubinovitz optimum combination of the multi-projects to minimize and Coffler (1996). In this case, the manufacturing the loss rate of material is an even more difficult problem. process of rebars produces few scraps with almost zero The optimization algorithm using computers is one of percent loss of raw materials. However, the generation the most effective ways to solve those problems. of waste is inevitable in countries which do not supply Previous research has shown the benefits of computer rebars in coils and even in capital-intensive countries applications to improve productivity, constructability, which can not supply rebars of larger than 16 mm safety and quality in the process of concrete diameter coils. reinforcement work. Bernold and Salim (1993) presented The straight bars of market lengths, normally 8.0, 8.5, placement-oriented design and delivery of reinforcement 9.0, 9.5, .., are produced in Korea, since the raw materials based on both computer integration and feature-based for rebars are not supplied in coils. The market length design concepts. They also proposed a concept of rebar production generates relatively many scraps after rebars delivery and staging based on a placement plan to are cutoff in required lengths. improve productivity on site (Salim and Berbold 1994). The waste rate of scraps increases when raw materials Dunston and Bernold (1994) produced a strategy for the are ordered without proper bar cutoff plans based on the robotic rebar bending based on experiments and structural review of drawings. The loss rate of scraps developed a control model for accurate rebar bending can be even higher as the diameter of rebars increases based on computer integrated manufacturing (CIM) (Kim 2002). The loss of materials can be reduced when concepts (Dunston and Bernold 2000). Navon et al. the most desired length of bars are ordered based on the (1995, 1996) described the benefits of computer-aided sufficient review of drawings and bar schedules. Extra design and computer-aided manufacturing (CAD/CAM) savings of rebars are possible when the special length of systems for concrete reinforcement and developed a a certain amount of tonnage is ordered to steel mill. model for rebar constructability diagnosis and correction in an object-oriented programming environment (Navon *Contact Author: Sun Kuk Kim, Associate Professor, College of et al. 2000). Architectural and Civil Engineering, Kyung Hee University,1 However, the direct approach to the optimization Seochon-ri, Kiheung, Yongin, Kyonggi-do, 449-701, Korea algorithm to reduce the loss rate of rebars was not found Tel: 82-31-201-2922 Fax: 82-31-203-0089 among these papers. The work carried out by Navon et E-mail: kimskuk@khu.ac.kr al. (1995) was one of the few studies which addressed (Received October 24, 2003 ; accepted April 6, 2004 ) the optimization algorithm for reducing the loss rate of Journal of Asian Architecture and Building Engineering/May 2004/23 17 steel rebars. However, Navon did not present the detailed The quality of labor provided by the subcontractor algorithm even though an optimization module based can significantly influence waste rate as well as rebar on linear integer programming (LP) solver-LINDO was works. Site investigation (Kim, 1987) shows that waste mentioned. of rebars decreases if optimum rebar combination and This paper, therefore, was prepared with the aim of systematic inventory management are properly carried developing algorithms to supply rebars required to out from ordering phase to manufacturing phase. The minimize material waste during cutting and bending of optimum combination of rebars, calculated by computer, discrete (single) bars in rebar shops. provides very useful information for the manufacturing of rebars as well as systematic inventory management Reasons for Loss of Rebars that reduces waste rate. The waste rate can be estimated as high as 3 to 5% in the bidding stage in countries where rebars are not Automatic Rebar Detailing Concept supplied in coils. This rate can be even higher than 10% The rebar combination process begins with the as the diameter of rebars increases. Kim (1987) showed preparation of Rebar Data Files (RDF) based on the that the loss rate from plant projects is higher than that structural calculation. Structural design data, however, of building construction in which rebars of typical length provides basic information related to arrangement of and diameter are repeated in drawings. Major causes rebars. Detailed rebar information including influencing the waste rate of rebars were identified from development and splice length, concrete cover and several management processes of rebar work as follows. interference of rebars is not expressed explicitly in (1) The highest rate of waste is observed when the structural drawings. Therefore, the automatic preparation purchase order with redundancy is made to steel mill of RDF from the information of structural design without accurate understanding of manufacturing requires a module that provides rebar detail. Fig 1 information, such as the structural drawings and bar graphically demonstrates the conceptual construction of schedules. The waste of materials rapidly increases when RDF. the proper attention is not paid to the surplus order of raw materials during the construction stage. Therefore, significant waste can be avoided if the required quantity of rebars is precisely analyzed and reflected in the mill order. (2) Materials are also wasted when surplus rebars with length of 2-3m are not reused after cutoffs. Better economy is achieved using rebars with margins of shorter than 1m without cutoff since the cost of labor related to cutting less than 1m is more expensive. The rebars of Fig.1. Automatic Rebar Detailing Concept extra length, with either straight or L shape, found from slabs and beams not only increase waste rate of materials After all structural design data of structural members but also add additional weight to the structure. Waste of including number of bars, diameters, geometric size of material as high as 1% can be saved when proper bars in each member, etc. are extracted from the structural market length are selected for combination in order not design data file (SDDF), splice and development length, to generate scraps of about 1m based on the structural concrete cover related to each structural member are, review of drawings (Kim, 1987). then, obtained from the structural member specification (3) It is also shown that approximately a 1% waste data file (MSDF). As a next step, rebar manufacturing rate occurs when cutting planning without consideration detail is prepared according to Automatic Rebar of bending margins is carried out. Detailing Algorithms (ARDA). (4) One of the frequent causes of waste is the failure The ARDA consist of two tasks. The first task is to of inventory management of rebars cut and bent. This automatically generate rebar details of all structural type of waste is observed in urgent and large-scale members, and the second task is to estimate precise construction projects. cutting lengths and quantities of rebars based on the (5) It is sometimes found from construction practice details obtained in the first task. Each structural member that the length and location of bar splices as well as needs several ARDA, depending on the arrangement developments are not observed in compliance with codes condition of structural members. For example, many to compensate for the loss of materials, since strict algorithms of beams are required for the estimation of application of codes can create significant loss of precise cutting length and quantity of rebars, depending materials. In these cases, therefore, the quality of rebar on bar types (bent bar, straight bar) and number of spans works is not satisfactorily controlled. (single span, and multiple). Kim and Kim (1994) (6) Inappropriate management of rebar shops and presented an automatic rebar detailing concept, and Kim layout of cutting and bending machines is another source (2002) proposed various detailing algorithms for all of waste of materials. structural members. 18 JAABE vol.3 no.1 May. 2004 Sun-Kuk Kim An Overview of Rebar Optimization The special order is initially considered for the bar The algorithm to reduce waste of rebars boils down combination process based on reading RDF prepared to the question of how efficiently scraps generated during by ARDA. At the same time, optimal conditions rebar work designated as structural drawings can be including waste rate (ε), quantity (q) and length (l) are minimized. For the multiple rebars as shown in Fig. 2, read to minimize rebar waste for special orders as shown the following steps are proposed for the algorithm in in Fig. 3. Rebars of lengths 7m, 8m and 9m, which can which l = combined length of rebars, L = length of rebars be easily purchased from the market by a normal order i i to be ordered, bar = rebars extracted from drawings, i, j (Normal order 1), are combined to match the rest of the = index, n = number of l or L , and ε = waste rate of rebars. The final combination of rebars with the market i i i combined bars. length of 6.0m, 6.5m, 7.5m, 10.0m, 11.0m and 12.0m (Normal order 2) is performed to produce rebars whose combinations were not found. The following are the explanations of Fig. 3. Fig.2. Example of rebar combination 1. Find l where 0 ≤ n (L- l )/L ≤ ε (1) i i i 2. Let sum = (2) The sum is to be multiplied by the unit weight of each size of bars. If sum ≥ tons where tons is the minimum quantity for order, then l (i=1, 2, .., m) can be selected for order and it is recorded into the resource field of bar . 3. Decrease the length gradually according to the given range. Fig.3. An overview of rebar combination process 4. Repeat the process 1 and 2 until L < Min (bar ). i j (1) Rebar arrays are prepared by sorting out rebars of Rebar Combination Process and Algorithms the same diameter based on readings of RDF prepared Even though the proposed algorithm looks simple, it by ARDA. The arrays include length, number of identical is difficult to find numerous combinations of rebars that bars and bar mark that retains the information of a bar satisfy given conditions including waste rate of material, type, diameter, spacing, and serial number. The RDF length and quantity for purchase order as well as structure is very similar to the one proposed by Navon, construction schedule. Computers can replace time- Rubinovitz, and Coffler (1995). consuming manual efforts, resulting in significant (2) The algorithm searches and combines rebars savings of materials, managing cost and labor. Material satisfying all the given conditions such as waste rate and waste can be minimized if the rebar combinations are quantity (for example, ε ≤ 1%, and q ≥300tons, obtained with not only market length but also special respectively) for rebar length specified by 6m ≤ S ≤ lengths with minimum quantity for order supplied by 12m (0.1m interval). First consideration is given to the steel mills. single bars for search since combinations for single bars The standard lengths of rebars that can be purchased are not necessary. The search is extended to combinations in Korean markets are 7m, 8m and 9m. Besides, It is that satisfy the given conditions with two bars. The search also required by Korean Standards that 6m, 6.5m, 7.5m, and combination are repeated with up to four bars. 10m, 11m and 12m be provided for the construction sites. (3) The combination with more than four bars is not If rebars are supplied by special order, waste can be attempted to avoid inefficient computing time since avoided further. For instance, if rebars of 320 tons with effective inventory management of rebar shops is both 8.7m length and 28mm diameter are delivered to difficult. It was observed from a sample project that sites by special order where the same quantity of 8.7m searching short rebars to be combined with primary rebars are required by structural design, the waste rate rebars of longer lengths minimizes computing time when is zero. This waste rate increases up to 3.3% with a scrap more than two rebars are combined. The binary search bar of 30 cm when 9m rebars are supplied by the normal algorithm, shown in Fig. 4, is adapted among various order. Construction costs increase rapidly when multiple searching methods to minimize time in finding rebars projects are carried out in rebar shops with poor to be combined based on given conditions from hundreds management systems during the ordering stage. of rebars sorted with respect to length. This binary search algorithm was proved by Horowitz and Sahni (1983). JAABE vol.3 no.1 May. 2004 Sun-Kuk Kim 19 algorithm is fast, even if the chance to minimize waste rate is lost. (2) Best-fit algorithm: This algorithm tries all possible combinations of every rebar stored in RDF; and a rebar combination that yields the least waste rate is selected. From the above example, the combination of a 7m rebar and a 3m rebar out of a possible six rebar combinations gives the least waste rate of zero while a waste rate of 2% calculated from 7m+2.8m rebar combination was obtained by the First-fit algorithm. This algorithm will not lose the chance to minimize waste rate, despite unavoidable computational time. (3) Modified-first-fit algorithm: This algorithm applies differentiated waste rate. Main rebars are combined within a specified waste rate (for example, waste rate of 1%) and a relieved condition waste rate, such as 3%, is applied to the rest of the rebars. However, the overall waste rate must not be beyond the target waste rate, such as 2%. It was shown from the algorithm test that quantities combined based on the relieved waste rate do not exceed 10% of total rebars under combination. This algorithm is considered to be the best algorithm empirically in maximizing combination efficiency and applicability while minimizing computing time. Fig.4. Binary search algorithm for rebar combination Algorithms for Mathematical Programming (4) The given conditions of e (ex. ε ≤ 3%) and 6m ≤ L It was shown that several algorithms must be combined ≤ 12m (for L = 7, 8, 9m) are, then, implemented to search to solve optimization problems minimizing rebar waste and combine rebars left over from Routine 1 (Normal rates. This study presented solutions to the following order 1). questions. (1) How can one prepare rebar-detailing (5) Routines 2 and 3 are followed by the combinations algorithms from the information of structural design? meeting the conditions of e (ex. ε ≤ 3%) and 6m ≤ L ≤ (2) How can one find a logical process for rebar 12m (for L = 6m, 6.5m, 7.5m, 10m, 11m, 12m) imposed combination? (3) What type of conditions must be on rebars left over from Routines 2 and 3 (Normal order imposed to combine rebars for each process (first or best- 2). Search for the rebar combination in Routines 3 and 4 fit algorithm)? (4) How can one rapidly search bar data also uses the algorithm shown in Fig. 4. to be used for rebar combination (binary search Three algorithms for the combination process as algorithm)? shown in Fig. 3 are presented in this study to solve the Besides, the problem as to how one can quickly optimization problem which reduces the waste rate of calculate rebar quantity satisfying given requirements rebars while taking computational time and practice into remains unanswered. Numerous calculations of rebar consideration. quantity are required to generate a candidate solution and numerous candidate solutions must be obtained to (1) First-fit algorithm: This algorithm combines rebars find an optimum solution. Algorithms from which were read from RDF based on the given waste mathematical programming are necessary to expedite rate (ε). The algorithm terminates the combining process such complicated calculations. Solving of the when waste rate calculated from the first rebar optimization of rebar work is described by linear combination are within the specified rate, in spite of programming. Algorithms based on Equations (1) and further possible reductions in waste rate with following (2) can be suggested using data read from RDF as a form combinations. For example, if rebars with of length 10m of array. is to be combined using rebars of 7m, 2.8m and 3.0m with the specified waste rate of 3%, the algorithm, based Decision variables on the order of rebars stored in RDF, finds the first P = total sum of combined rebars for all structural combination of 7m+2.8m=9.8m with waste rate of 2% members (((10-9.8)/10)*100). The waste rate calculated from this Q = total sum of rebars to be ordered for all structural combination is less than the given waste rate of 3%. The members combination of 7m long rebars with 3m rebars instead p , …, p = sum of rebars combined in a certain length, 1 n of 2.8m rebars is not considered, even though the waste q , …, q = sum of rebars to be ordered in a certain length 1 n rate from combining with 3m rebars is zero. This 20 JAABE vol.3 no.1 May. 2004 Sun-Kuk Kim st th Objective Function structural members of all floors (1 -20 floor) of 5 buildings except the foundation. Table 1 is a combination list for the special order (combination number S1) of Minimize (3) 10.6m long rebar. Rebars are combined with the maximum material loss (waste rate) of 1.0% and minimum quantity of 300 tons. or (4) Table 1. Combination List for Special Order Subject to (5) (6) Where the weight (tons) is the minimum quantity of a purchase order, and ε represents the waste rate defined by users. Decision variables, P, p , Q, q , of Equations i i (3) and (4) are calculated by Equations (7), (8), (9) and (10). (7) Where w is the unit weight of the rebar. One rebar element, p out of all rebars combined within a certain i, length, {p , .., p }, is expressed as Equation (8). 1 m (8) Table 2. Combination List for Normal Order Where a is the number of rebars combined within a certain length and x is the actual length of combined rebars. The quantity of rebars to be ordered in a certain length is given in Equation (9). (9) One rebar element, q , out of all rebars to be ordered in a certain length, {q , .., q }, is obtained from Equation 1 m (10). (10) Where y is the market length of rebars to be ordered. Test Results of Proposed Algorithms The algorithm presented in this study was programmed with Visual C++ 4.0. Both the accuracy and applicability D: Diameter in mm, L:Length in m, B.M.: Bar Mark, Nos: Number of bars’ under Table 2 to sites of the theory was validated by a case study of high-rise residential buildings described as follows. - Location : Yongin-si, Kyonggi-do, Korea The modified-first-fit algorithm was used to - Total floor area : 92,435 m (20 stories x 5 buildings) economically combine rebars designated as in the first column of Table 1. These rebars meet specified - Structure : Bearing wall system combining conditions. Single rebar lists without the need - Duration of main structural work : June, 2000 - of bar combination are shown by S1-1 through S1-10. December, 2000 (6 months except foundation work) Five buildings managed by the same schedule of S1-11 is the list of special orders for combinations with reinforcement work were considered for system tests in two rebars. For example, the waste rate is zero since the this study even if the complex consisted of 20 buildings. rebars represented by S1-1 and S1-2 is 10.6m. S1-10 Tables 1 and 2 are the results of the combination for the generates a waste rate of 0.94 % and is included in the 16mm rebar most frequently used in the project. Every list. S1-11, which combines Bars 0085 and 0054, satisfied rebar was assumed to be systematically managed during the condition of special order with waste rate of 0.28 %. the 6 month construction period of the main structural The modified-first-fit algorithm, which selects long rebars, is followed by the combination of short rebars to framework. The combination was performed for JAABE vol.3 no.1 May. 2004 Sun-Kuk Kim 21 reduce computing time. All Bars 0085 are used for S-11 The difference of 40mm in length between Table 3 combination, whereas only 660 bars from bars 0054 (the and Fig. 5, called bending margin, resulted from the total of 1645 bars) are used for the combination of S1- increase during bending process. In case of deformed 11 (Table 3). The rest of bars 0054 are used to combine bars, the bending margin is calculated by 2.5 times of S1-24, S3-5, N1-18, N15-3 as shown in Table 1. the rebar size in diameter. Finally, Table 1, containing results of the combination of 23,945 rebars of length 10.6m, compares the order weight of 395.955 tons with the net weight of 393.702 tons, demonstrating a waste rate of 0.469%. The real diameter of the nominal 16mm diameter rebar is 15.9mm while the unit weight is 1.56kg/m in KS (Korean Standard). Rebar notations are defined by T16, H16 and B16 instead of Y16 to clearly indicate the diameters and locations of rebars. T, B, H, V and M represent top, bottom, horizontal, vertical and middle bar, respectively. Table 2 shows the combination list for normal orders Fig. 5. A sample of bar tag (combination number N1) of 9.0m long rebars. For this combination, the maximum loss is 1% and there is no Table 4 are the results of rebar combination for special limit to the minimum quantity for order. Rebars of 7, 8, orders (S1, S2, S3, …) and normal orders (N1, N2,…) and 9m are combined first according to the algorithm based on given requirements, representing information employed in this study and the combination based on including rebar length, number of rebars, weight of each rebars of 6.0m, 6.5m, 7.5m, 10.0m, 11.0m and 12.0m is combination. followed. However, the execution of combinations of Table 4. Report of Combination Results 11m and 12m rebars is delayed till the order from site due to the traffic condition. Table 3 represents rebar sources of combination managing all combined rebars by bar number. It provides systematic tools for inventory management of rebar cut off, enabling a bar bending process quickly and effectively. Without these lists, it is difficult to locate combined rebars for bar bending even if the rebar combinations are successfully performed. For example, since rebars 0054 are used for the combinations of S1- 11, S1-24, S3-5, N1-18, N15-3, one must keep track of each combination to locate 1,645 rebars. Table 3 demonstrates results of all combined rebars, while rebars of the same diameter can be printed out if necessary. The detailed manufacturing information of Table 3 which can be checked from the rebar schedule and bar tag as shown in Fig. 5 is used to facilitate inventory management of manufactured rebars. For instances, S1 is related to special purchase order, one piece of information obtained from combination Table 3. Rebar Sources of Combination results shown in Table 1. Table 4, therefore, is used for order of rebars manufactured according to combination lists. It was shown from the combination of 16mm of the sample project that total order quantity was 2,937.265 tons with the waste rate of 0.819%. The reason that the total waste rate of 0.819 is larger than those of S1 (0.469) and N1 (0.392) shown in Table 1 and 2 respectively is due to the larger waste rate of the rest of the rebar combinations caused by the relieved condition of the modified-first-fit algorithm. The waste rate obtained from the secondary combination of rebars left over from the main combination is calculated to be over 3%. However, the influence of the secondary combination on the overall waste rate is insignificant because the rebar influenced by the secondary combination is relatively small compared to the total 22 JAABE vol.3 no.1 May. 2004 Sun-Kuk Kim quantity. The actual construction data revealed that the structural design data was a difficult task in this study. project used 3,007.97 tons of 16mm rebar, generating A more efficient format, automatically using structural the waste rate of 3.246%. The waste rate could have been design data for the management of rebar work, needs to reduced down to 0.819%, saving 70.705 tons of rebar be developed in order to effectively deal with a series of (2.427% waste rate) equivalent to 30,300 USD (about works including structural design, the preparation of 35 million Korean won) if the algorithm proposed by structural drawings and rebar details, and optimization this study had been implemented for the construction of rebar work. management of rebar work. If the construction of all 20 buildings under consideration had been carried out References 1) Bernold, L.E., and Salim, Md. (1993) “Placement-oriented design applying the algorithm of this paper, approximately 400 and delivery of concrete reinforcement.” J. Constr. Engrg. and tons of steel worth 170,000 USD (199 million Korean Mgmt., ASCE, 119 (2), 323-335 won) could have been saved. 2) Dunston, P.S., and Bernold, L.E. (1994) “Adaptive control for robotic rebar bending.” Microcomputers in Civ. Engrg., Oxford, Conclusions England, 9, 53-60 3) Dunston, P.S., and Bernold, L.E. (2000) “Adaptive control for safe A Considerable waste rate occurs if sufficient attention and quality rebar fabrication.” J. Constr. Engrg and Mgmt, ASCE, is not paid to the management of complex rebar work of 126 (2), 122-129 construction projects. A great deal of rebars can be saved 5) Horowitz, Ellis, and Sahni, Sartaj (1983) “Fundamentals of data with increased productivity when purchase orders, structures”, Computer Science Press, Inc., Maryland, 342 manufactures and installations are carried out according 6) Kim, S.K. (1987). “A Report of Rebar Waste Rate Analysis of RC Structures”, Daelim Industrial Co., Ltd., Korea, 16-17 to construction schedules, while both rebar details and 7) Kim, S.K., and Kim, C.K. (1994) “Integrated Automation of required optimal rebar quantities are prepared based on Structural Design and Rebar Work in RC Structure”, J. Structure the algorithm presented in this study. and Construction, the Architectural Institute of Korea, 10(1), 113- The example run demonstrated the reduction in the 8) Kim, S.K. (2002) “A System Development for Automatic Detail waste rate by about 2.4 percentage points by st Design and Estimation of Rebar Work”, the 1 year Research implementing the algorithm of this paper. Relatively Report, Gyeonggi Regional Small & Medium Business much reduction of waste is expected from plant Administration, Korea, 84-90 construction involved with various types of rebars than 9) Navon, R., Rubinovitz, Y., and Coffler, M. (1995) “RCCS: Rebar from the construction of high rise residential and CAD/CAM System” Microcomputers in Civ. Engrg., Oxford, England, 10, 385-400 commercial buildings in which rebars of typical length 10) Navon, R., Rubinovitz, Y., and Coffler, M. (1996) “Fully automated and diameter are repeated throughout design. rebar CAD/CAM system: economic evaluation and field The example study also demonstrated the combination implementation.” J. Constr. Engrg and Mgmt, ASCE, 122 (2), 101- algorithm, among all algorithms presented in this research, enhances not only computing efficiency but 11) Navon, R., Shapira, A., and Shechori, Y. (2000) “Automated rebar constructability diagnosis.” J. Constr. Engrg and Mgmt, ASCE, 126 also the efficiency of rebar management related to (5), 389-397 construction schedules. The modified-first-fit algorithm 12) Salim, M., and Bernold, L.E. (1994) “Effects of design-integrated is considered to be the best algorithm empirically in process planning on productivity in rebar placement.” J. Constr. maximizing combination efficiency and applicability Engrg. and Mgmt., ASCE, 120 (4), 720-738 while minimizing computing time. The prompt extraction of precise rebar data from JAABE vol.3 no.1 May. 2004 Sun-Kuk Kim 23

Journal of Asian Architecture and Building Engineering – Taylor & Francis

**Published: ** May 1, 2004

**Keywords: **rebar work; optimization; algorithm; waste rate; combination

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