Journal of Constructional Steel Research 67 (2011) 1435–1441

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Journal of Constructional Steel Research

Optimization of Off-Centre bracing system using Genetic Algorithm

H.A. Mosalman Yazdi a,⁎, N.H. Ramli Sulong b

a Civil Engineering Department, Faculty of Engineering, Islamic Azad University-Maybod Branch, Iranb Civil Engineering Department, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia

⁎ Corresponding author. Tel.: +98 9131516522.E-mail addresses: hmosalman@um.edu.my, hmosalm

0143-974X/$ – see front matter © 2011 Elsevier Ltd. Adoi:10.1016/j.jcsr.2011.03.017

a b s t r a c t

a r t i c l e i n f o

Article history:Received 14 November 2010Accepted 16 March 2011Available online 5 May 2011

Keywords:Off-Centre Bracing systemBucklingGenetic algorithm

In this paper, a method based on genetic algorithm is proposed for determining the optimum connectionpoint with the highest lateral buckling load in the Off-Centre bracing system. This type of bracing system ismostly used in seismic areas and it allows architects to have more openings in the panel area. In this system,the non-straight diagonal member introduces eccentricity to the system and is connected to the corner of theframe by a third member. In designing this system, designers often use “trial and error” to locate theconnection point of the brace elements considering various parameters affecting the design such as openingand frame dimensions, cross sectional areas of brace elements and the location of the brace elementconnection. Hence, finding the best connection point with maximum lateral buckling load can be problematicby the conventional methods. To demonstrate the effectiveness of the proposed GA method, examples withdifferent frame specifications were presented.

an@gmail.com (H.A.M. Yazdi).

ll rights reserved.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Optimization is a challenging process in industrial design aswell asstructural design. By optimization, designers are able to save time andmoney with better design. However, there are many differenteffective criteria in the structural optimization, for which the use ofconventional optimizationmethods such as mathematical approachesis quite difficult. Since the early 1960s using different methods basedon human evolution has started [1]. This class of methods, calledevolutionary computation, is used for complex optimization tasks.One of these approaches is Genetic Algorithms (GA), which is anoptimization and search technique based on the principles of geneticsand natural selection [2,3]. Although genetic algorithms were firstinitiated by Holland in the 1970s [4], they attracted significantattention in a number of fields as a methodology for optimization,adaptation, and learning after the publication of Goldberg's book in1989 [5].

The application of GA's methods in the Civil Engineering field hasbeen developed in recent years, such as in construction management[6]. The application of genetic algorithms has expanded in variousfields of optimization in structural engineering [7,8] as well as intopology optimization [9–12] and geometrical specification ofstructural elements [13–15]. In this paper, GA is used for optimizationof the eccentricity and determination of the optimum midpointconnection of a particular type of braced system.

Generally, braced frames are one of the most common types forresisting structures against lateral loads. Braced elements are dividedinto two groups: concentric and eccentric [16–19]. Easy construction,relative good stiffness and economical aspects make concentricbraced systems more desirable than eccentric ones [18–20]. Eccentricbraced systems have higher energy dissipation and are more durableagainst the seismic loads in spite of the higher cost and constructionaccuracy required [21,22]. In order to overcome the shortcoming ofthese bracings, particularly the concentric braced system, some othertypes of bracing, such as the knee bracing system, have beendeveloped [23,24]. In this paper, a particular type of braced frame,called Off-Centre Braced (OCB) system, is introduced. This system ispreferred by architects because it provides a greater possibility ofopenings in the panel area [25]. This braced system, as shown in Fig. 1,consists of three members in which the third member connects themid connection point of the brace elements to the frame corner. Thecyclic nature of the earthquake load causes designers to apply adouble bay rather than a single bay for this system (see Fig. 2).

Although, a few researchers had investigated this system undertensile brace elements [25,26], evidence from recent earthquakes hasshown that out-of-plane buckling of the mid connection (point O) isdue to compressive brace elements, which leads to severe damage ofstructures. Guidelines for determining the best member configurationfor this system are not available.

Consequently, determining the mid connection location with thehighest lateral buckling load capacity, considering effective parame-ters such as mid connection location, bracing cross sectional areas anddimensions of frame and opening are important. Therefore, tooptimize the eccentricity of the mid connection (or in other wordsfinding the optimum connection point) a method based on genetic

Fig. 1. An Off-Centre braced system.

Fig. 3. A model of OCB frame.

1436 H.A.M. Yazdi, N.H.R. Sulong / Journal of Constructional Steel Research 67 (2011) 1435–1441

algorithms is proposed to facilitate the design procedure of thissystem. In this respect, investigation of certain operators such asselection, crossover, mutation and elitism has been carried out toprovide a proper design. By comparing the GA results withconventional method, it has proved that the selected connectionpoint by the GA approach is one of the best connection points.

2. Theoretical calculations of buckling load

The behavior of OCB systems against lateral load is one of the mostimportant characteristics of these systems. As shown in Fig. 2, byimposing a lateral load to this braced frame, the three brace members,which are connected at point O, will be in compression in one bay, andthe three members in the other bay will be in tension. Under cyclicloading, the member forces in these two bays will be changed (vice-versa) in the subsequent cycle.

In analyzing this system, the Off-Centre Braced (OCB) frame, asshown in Fig. 3, is assumed as a truss system, where the bracingelements are connected at point O. The dimensions of the memberscan be defined by two parametersm and n, which are the coefficientsof the span and height of the frame, respectively.

With the above definition, the axial forces of these bracingmembers can be obtained as follows:

F1 =−PHLSinα1

ð1Þ

F2 =−P

Sinα2ð2Þ

F3 =−P

Sinα3

Cosα2

Sinα2−H

L

� �ð3Þ

where α1, α2, and α3 are the angles between the bracing elements 1,2 and 3 to the beam and column with respective lengths of L1, L2 andL3.

The lateral buckling load of this system depends on the openingdimensions, section properties of the frame elements and the type of

Fig. 2. Axial forces in OCB frame due to applied lateral load, P.

mid connection. In this research, the mid-connection is consideredrigid and the members are considered pinned at their endconnections.

To determine the moments and forces at both ends of the braceelements, as shown in Fig. 4 a displaced beam-column element withits end moments and axial forces is taken.

Then, the Slope-Deflection method is used to obtain the momentsand shear forces at both ends of elements. Finally, by imposing theboundary conditions of the brace element's connections, the equa-tions are simplified [27,28]:

M1 =EIl

Sθ1 + Scθ2ð Þ ð4Þ

M2 =EIl

Scθ1 + Sθ2ð Þ ð5Þ

Vl = − M1 + M2ð Þ−Fϕl ð6Þ

Fig. 4. Consideration of both ends rotation along with relative lateral displacement.

1437H.A.M. Yazdi, N.H.R. Sulong / Journal of Constructional Steel Research 67 (2011) 1435–1441

ϕ =Δl

ð7Þ

where M1 and M2 are the end moments of each element byconsidering the slopes of the members θ1and θ2. V, F,E,Iand l are theshear force, axial force, module of elasticity, moment of inertia andlength of the element, respectively. Δ is the support displacementperpendicular to the frame plane. In the above equations, S and c arestability functions, which are computed as follows:

S =α sinα−α cosαð Þ2−2 cosα−α sinα

ð8Þ

c =α− sinα

sinα−α cosαð9Þ

α2 =Pl2

EI=

π2PPe

= π2ρ ð10Þ

where, Pe, the Euler buckling load of this beam-column is:

Pe =π2EIl2

ð11Þ

Based on the aforementioned equations, the lateral buckling loadof the OCB system with rigid mid connection of OCB system can bedetermined by applying the slope-deflection method at the midconnection.

In order to use the equilibrium equation at this connection point,an origin is taken for this point to determine the elements' angle, inwhich O1, O2and O3 are the angles of elements 1, 2 and 3, respectively,as shown in Fig. 3.

The moments, MOi, and shear forces, VOi, of the brace elements,i=1,2,3 at the connection point, O, by considering boundaryconditions are (see Fig. 3):

Ki =EiIiLi

ð12Þ

MOi = KiS″i θOi−ϕið Þ ð13Þ

VOili = Ki −S″iθOi + S″i−π2ρi� �

ϕi

� �: ð14Þ

In out-of-plane direction, the slope θ for each element at theconnection point can be introduced by θx and θy. Based on thecompatibility, with consideration of the angles of the brace elementsto the origin at point O (see Fig. 3) we have:

MOi = KiS″i θx cosOi + θy sinOi−ϕi

� �: ð15Þ

Considering the equilibrium at the mid-connection point O,where:

∑3

i=1MOi = 0 ð16Þ

∑3

i=1Vi = 0: ð17Þ

The stiffness matrix for this system is as follows:

a11 a12 a13a21 a22 a23a31 a32 a33

24

35 θx

θyΔ

24

35 = 0 ð18Þ

where,

a11 = ∑3

i=1KiS

″i cos

2Oi ð19Þ

a12 = a21 = ∑3

i=1KiS

″i cosOi sinOi ð20Þ

a13 = a31 = − ∑3

i=1

KiS″i cos

2Oi

lið21Þ

a22 = ∑3

i=1KiS

″i sin

2Oi ð22Þ

a23 = a32 = − ∑3

i=1

KiS″i sin

2Oi

lið23Þ

a33 = ∑3

i=1

Ki S″i−π2ρi� �

li: ð24Þ

The stability of this system in out of the plane depends on thebuckling of the brace elements. Thereby, the lateral buckling load of thissystemcanbeobtainedby taking thedeterminant of thismatrix as equalto zero. This critical load is very useful for designers to determine theeffective length coefficient. A MATLAB program is developed based onthe above equations for determination of the lateral buckling load.

3. Genetic algorithm

The Genetic Algorithms (GAs) are search and optimization algo-rithms that are based on the Darwinian theory of evolution. Theunderlying concept is that reproduction is based on the rule of “survivalof the fittest” [1,29,30].

The Genetic Algorithm is started with a set of solutions called thepopulation. These solutions are represented by chromosomes that arestrings of genes. Solutions from one population are taken and used toform a new population, with expectations of having a better newpopulation than the old one. Solutions are selected to form newsolutions, so-called offspring, which are selected according to theirfitness. The one with more fitness value has a better chance ofselection in the reproduction procedure. This is repeated until somecondition is satisfied. Some of these termination criteria, as taken inthis investigation, are the number of generations and the maximumfitness value [31,32].

Generally, one of the most important concepts of the geneticalgorithm is the fitness function. A fitness function is a particular type ofobjective function that quantifies the optimality of a solution (that is, achromosome or an individual in the population) so that a particularchromosome may be ranked against all the other chromosomes.Intuitively, we can think of the fitness function as some measure ofprofit, utility, or goodness that we want to maximize. The fitness of anindividual is related to its influence upon the future development of thepopulation. In this research the lateral buckling load of the system istaken as thefitness function. Also, GA is applied to obtain one of the bestmid connection points with the maximum lateral buckling load [3,29].

3.1. GA operators

Generally, the individuals of each generation inGA are represented byinteger strings. Each string encodes apossible solution to theproblem. It isstarted from an initial population that consists of randomly selectedstrings. The search for the best solution proceeds through the generalprocess of reproduction. New strings are created and replace less fitsolutions in the present population. From generation to generation, this

Fig. 6. Flowchart of genetic algorithm for finding the best connection point of braceelements.

1438 H.A.M. Yazdi, N.H.R. Sulong / Journal of Constructional Steel Research 67 (2011) 1435–1441

leads to a higher overall fitness of population and to better solutions[33,34].

The reproduction component of GAs consists of three geneticoperators: selection, crossover, and mutation. The selection operatorselects a string from the population to do reproduction. In general,strings are selected such that those with high-fitness have a higherchance of being selected. In this research, Ranked-based selection isadopted for the selection scheme. Rank-based selection does notcalculate the selection probabilities from the fitness values directly. Atfirst, it sorts all individuals according to their fitness values and thencomputes selection probabilities according to their rank rather thantheir fitness values [3,35].

The crossover operator combines two selected strings to produce apair of new solutions. Usually a crossover point is randomly chosen atwhich the two parent strings are cut. After parts of both strings havebeen interchanged, they are “joined” together to produce two newstrings (see Fig. 5) [3,36].

The mutation operator adds a random component to the GAs.Mutation randomly changes the integer components of a string, andthe mutation of each component takes place according to a pre-selected threshold referred to as the mutation rate or mutationpossibility. For example, if the mutation rate is 0.1, a random numberis generated for every component in the string by a uniform randomnumber generator [between 0 and 1] (see Fig. 5).

This operation can bring about a solution not present in thepopulation, or protect against accidental loss of valuable informationdue to, for example, unfavorable crossover. The mutation probabilityof an integer is usually chosen to be low, in order to minimizeinterference with the main genetic operations [30,37].

In this proposed GA method, the elitism strategy is used in theprocedure of optimization. In other words, fitness-proportional selectiondoes not guarantee the selection of any particular individual, includingthe fittest, except it is guaranteed to be between the selected individuals.Thus, with fitness-proportional selection the best solution to the problemdiscovered so far can be regularly discounted. Thereby, ensuring thepropagation of the elite member, which is called elitism, and which is tobe retainedwithin the individuals for the next generation. This individualwill not be disrupted by crossover or mutation [37,38].

3.2. Development of GA program

For finding the best connection point another MATLAB programadopting genetic algorithm is developed. In order to find the bestbraced frame connection point, which has the highest lateral bucklingload, two parameters, m and n, are introduced as unknownparameters and all brace element cross sections are taken the same,because of practical consideration in the construction of structures.Although, the program is capable to consider various design variablessuch as brace element cross section area, frame rigidity, braceconnection, frame and opening dimensions. Hence, based on theaforementioned explanation the GA procedures can be summarized inthe flowchart shown in Fig. 6.

Fig. 5. Crossover and m

Randomly generated individuals in each generation that indicatethe mid connection location, have the possibility of being out of thefeasible area. Therefore, to decrease the iteration number ofgenerations and to avoid unfeasible individuals between populations,a feasible area based on the dimensions of the opening and frame isintroduced. There are many points out of the opening region in thepanel area that have a high lateral buckling load but are not suitable tobe a location for the brace element connection. This is because thepoint might obstruct the construction of the opening area. In thisrespect, the feasible area for the braced frame is defined as thecommon area between the beam, columns and the two lines that startfrom the corner of the frame (point B and C) pass through the cornerof the opening and end at the beam and column (point M and N), asillustrated in Fig. 7. Therefore, by determining this area (which ishatched in Fig. 7) a good filtration can be performed and eventually

utation operators.

Fig. 7. Possible panel area for locating brace element connection.

1439H.A.M. Yazdi, N.H.R. Sulong / Journal of Constructional Steel Research 67 (2011) 1435–1441

prevent utilization of impossible individuals with respect to theirlocation for better and faster convergence.

4. GA investigation

In this section, the accuracy of the proposed approach based ongenetic algorithm in determining the optimum connection point withthe highest buckling load is compared against a conventional method(based on trial and error) by using Eq. (14). For this investigation, fourOCB frames with different dimensions (H/L) and opening sizes (h/l)are adopted. The input data of the GA proposed program are shown inTable 1.

After running the MATLAB program, the result proposes aconnection point for each frame that is located from the top cornerof the framewith distances of X and Y. The results of frames are shownin Table 1. Input and output data of GA program are shown in Table 2.For instance, the output of third example introduces the connectionpoint with X=53 cm and Y=40 cm, which is referred to as m=0.4and n=0.4. This point, as the best selection, is obtained after 191iterations. The critical lateral buckling load of this system with theproposed location of connection point is 6.549 tons.

Table 1Input and output data of GA program.

Input data

Example no. 1 2 3 4

H/L (m/m) 400/400 300/400 100/133 100/133h/l (m/m) 2.5/2 2.7/1 60/80 60/93Brace element section size IPE140 IPE140 L50×50×6 L50×50×6Brace element's momentof inertia I (cm4)

541 541 12.8 12.8

Selection rate (%) 50 50 50 50Module of elasticity E kg/cm2 2.1e6 2.1e6 2.1e6 2.1e6Mutation rate (%) 5 5 4 4Number of bits (gene) 32 32 16 16Number of population 100 100 60 60Maximum of iteration 500 500 300 300

Table 2Output data of GABUCKLING program for example.

Output

Example no. 1 2 3 4

X (cm) 266.31 299.98 52.9849 93.96Y (cm) 67.1 30 40.006 39.98Iteration no. 249 142 191 211Critical lateral buckling load (ton) 23.335 33.761 6.549 5.727

4.1. Mutation rate investigation

In this section, themutation rate or mutation probability regardingthe GA program is investigated. This program involves a large numberof calculations because of different parameters, such as, number ofiterations, maximum cost function for termination, number of bits,number of individuals and checking the individuals to be in thefeasible area. Therefore, obtaining results with a lower number ofiterations is important. One of the most important factors in thisregard is the mutation rate. An improper value of the mutation ratecan lead to early convergence, in which the result might not be thebest selection. To examine the effect of the mutation rate, a new OCBframe with H/L=4/4 and opening dimensions of 2.5×2 is taken. Allthe characteristics of this investigation except their mutation rate areconsidered the same. The mutation rates (Pm) are taken as 0, 1 and10%. As indicated in Fig. 8, without considering mutation, a prematureconvergence happens and the GA is not able to investigate all possiblepoints in the feasible area. For the case of a 1% mutation rate, bychanging bits and creating new individuals, other points in thefeasible area are evaluated by the fitness value. Therefore, byimporting some random individuals in the population, the probabilityof finding better selection in the feasible area increases (see Fig. 8).

For a mutation rate equal to zero, as shown in Fig. 8, the programconverges quickly. In this example, the best selection is obtained atgeneration number 6 and the maximum critical lateral buckling loadis 22676 kg (22.7 tons). With a mutation rate equal to 1%, the bestselection is obtained at generation 35 with a maximum lateralbuckling load of 23199 kg (23.2 tons) (see Fig. 8). Finally, for amutation rate equal to 10%, themaximum critical lateral buckling loadfor the selected connection point is 23336 kg (23.3 tons), which isobtained at generation 250. As indicated in Fig. 8, the variation ofaverage lateral buckling load in this case is more than the previouscase with a 1% mutation rate. This is because of more randomindividuals in each generation, which were created by the 10%mutation rate.

To investigate the effect of the number of individuals andmutationrates, two frames are examined. The first frame has dimensions of H/L=4/4 and an opening of 2.5×2 m. The characteristics of the secondframe are: H/L=3/4 m and its opening is 2.7×1 m. For both of theseframes the moment of inertia of brace elements is taken as 541 cm4

(IPE 140). The GA program is used to determine the best connectionpoint in the feasible area with the maximum critical lateral bucklingload. In this comparison, the number of individuals (population) ineach generation and the mutation rate (Pm) are varied. Thesecalculations are done based on the number of bits=16 and theselection rate=50%.

Fig. 8. Lateral buckling load for different mutation rate (Pm) for frame H/L=4/4.

Table 3Lateral buckling load for some points in the feasible area of example no. 1 (ton).

m 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.1 6.19 7.64 8.88 10.62 13.2 17.07 21.930.2 7.64 11.16 13.39 15.87 18.94 22.210.3 8.88 13.39 16.16 18.88 21.700.4 10.62 15.87 18.88 21.480.5 13.20 18.94 21.700.6 17.07 22.21

Infeasible area

n

Table 4Lateral buckling load for some points in the feasible area of example no. 2 (ton).

m 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.1 7.4 8.36 9.7 11.7 14.84 20.05 28.740.2 12.62 14.76 17.12 20.56 25.530.3 16.02 19.43

Infeasible area

n

Fig. 9. Effect of mutation rate on the average of lateral buckling load in the generation.Fig. 10. Mutation rate-population size variations on the best individual.

1440 H.A.M. Yazdi, N.H.R. Sulong / Journal of Constructional Steel Research 67 (2011) 1435–1441

As illustrated in Fig. 9, when the mutation rate is equal to zero, theaverages of fitness values do not have a stable condition. Thisinstability is due to premature convergence. In other words,populations in a local or restricted area are converged and a smallnumber of points in the feasible area do not have this opportunity tobe evaluated in the program. As indicated in these graphs, byimposing the mutation rate, the opportunity for the new points (orindividuals) in the population of each generation to be evaluated canbe increased. Therefore, themutation rate prevents local convergence.Therefore, the average fitness values (or average of the lateralbuckling load) in each generation with increasing mutation rate willbe decreased. With increasing mutation rate, the percentage ofrandom population increases (see Fig. 9). Based on the aboveexplanations and their results, the best mutation range for the GAprogram is proposed to be between 1 and 5%. The selection of a propernumber of populations in each generation can affect the output of theGA program. As shown in Fig. 10, except for a mutation rate equal tozero (which does not have a stable condition), the results do not havesuch a significant variation with different mutation rates. However, itcan be observed for population sizes less than 60 individuals alongwith mutation rates of less than 2%, that outputs may not be the bestselection. Therefore, it is recommended that the number of popula-tions should be at least 60 individuals.

5. Results comparisons

Further investigation on the effectiveness of the proposed GAmethod is done by comparing the results with conventional method.To examine the output of GA program and its selection, someconnection points in the feasible area, which are introduced bycoefficient m and n are selected.

The critical lateral buckling loads of these selected points of examplesare calculated based on Eq. (14) and are shown in Tables 3–5. Asillustrated in these tables, all the possible connection points in the feasiblearea have a lower critical lateral buckling load in comparison to the

selected point by the GA programwhich is indicated in Table 1. Input andoutput data of GA program are indicated in Table 2. Hence, it can beconcluded that the determined locations by the proposed GAmethod arewith the highest lateral buckling load capacity among the all connectionpoints in the feasible area. Consequently, it can be concluded that theselected points for the brace element connection, by this GA program arevery precise.

Finally, by determining the best connection point, relatedparameters for designing brace elements are calculated by MATLABprogram. For instance, two frames as named with example nos. 3 and

Table 5Lateral buckling load for some points in the feasible area of example no. 3 (ton).

m 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.1 1.582 1.789 2.074 2.502 3.174 4.2880.2 2.695 3.153 3.659 4.392 5.4530.3 3.417 4.149 4.797 5.6410.4 4.115 5.023 5.741 6.5480.5 4.95 5.8960.6 5.930

Infeasible area

n

Table 6Input and output data for MATLAB program.

Input data

Example No. 3 4

H/L=100/133, I =12.8 cm4 m and n X=52.98,Y=40.00

X=93.96,Y=39.98

E = 2:16 × kg = cm2 m=0.4, n=0.4 m=0.3, n=0.4

Output

Brace element no. i 1 2 3 1 2 3

Length (cm) Li 89.42 80.26 66.65 101.5 72.1 56.5Axial force (ton) Fi 10.98 9.86 4.09 10.90 10.32 6.07Euler bucklingload (ton)

Pei 33.17 41.17 59.71 25.74 51.02 82.92

Effective lengthcoefficient

Ki 1.73 2.043 3.81 1.53 2.22 3.69

Critical lateral bucklingload (ton)

6.548 5.727

1441H.A.M. Yazdi, N.H.R. Sulong / Journal of Constructional Steel Research 67 (2011) 1435–1441

4 are taken for calculations. The program inputs and output are shownin Table 6, where the required loads and parameters for designingbrace elements, such as axial load, Euler buckling load, effective lengthcoefficient and critical buckling load can be obtained.

6. Conclusion

The Off-Centre braced system is a special type of steel braced framethat is preferredbydesigners becauseof increasedpossibility formakingopenings. Determination of the mid connection location with consid-eration of some effective parameters, such as brace element crosssection, opening and frame dimensions, is problematic with conven-tional methods. Hence, in this paper a method based on geneticalgorithm is proposed to determine the best connection with highestlateral buckling load in the feasible area. In this regard, a MATLABprogram is developed to do the calculation. Comparisons withconventional method for different frame specifications were presented.The results showed that the proposedmethod is capable of determiningthe best connection point and its lateral buckling load with goodaccuracy.Moreover, investigations using the proposed GAprogram leadto certain recommendations for better and faster convergence of theprogram, suchas, themutation rate is better to bebetween1and5%, andthe number of individuals in each generation should be at least 60.Finally, it could be concluded that the proposed GAmethod is useful fordesigners to determine one of the best connection points in the feasiblearea.

Acknowledgment

It is acknowledged that the study is supported by the University ofMalaya and Ministry of Science, Technology and Innovation (MOSTI)Malaysia.

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