WebsJournals Epistemics in Science, Engineering and Technology, Vol. 2, No. 1, 2012, 43-50 Epistemics in Science, Engineering and Technology, Vol. 2, No. 1, 2012, 43-50
Epistemics in Science, Engineering and Technology Anwar et al
Genetic Algorithm Based Design of Bearing Thrust Block for Horizontally Bent Ductile Iron Pipes A.R. Anwar1, A.Y. Abdul Fatah2, O.A.U. Uche3, A.A. Adedeji4 Department of Civil Engineering, University of Ilorin, Ilorin, Nigeria
[email protected] 2 Department of Civil Engineering, Bayero University Kano, Kano, Nigeria
[email protected] 3 Department of Civil Engineering, Bayero University Kano, Kano, Nigeria
[email protected] 4 Department of Civil Engineering, University of Ilorin, Ilorin, Nigeria
[email protected] 1
Abstract The paper is aimed at optimizing bearing thrust block design for thrust restraint using Genetic Algorithm. The design was achieved by calculating the bearing Area from the design equation and then breaking the area in to the corresponding bearing width and height (b and h). The optimization was achieved by obtaining optimum values of the design variables (bearing width and height) that would give minimum volume of the truncated pyramid concrete block at the minimum bearing Area.Optimum values of the design variables obtained from genetic algorithm were successfully compared with the different values of the design variables obtained using the traditional Iterative approach. Five different iterates were compared with the Genetic Algorithm result for different soils. The volume was found to increase up to 95%, 90.97%, 74.55%, 59.72%, 33.77% and 119.05% for soft clay, silt, sandy silt, sand, sandy clay and hard clay respectively in contrast to Genetic Algorithm result. Optimum solution was found when the block height (h) varied between one and two times the block width (b) except for hard clay in which the block width was between one and two times the block height (h). Keywords: Genetic Algorithm, Thrust Force, Thrust Block, Optimization
1.
Introduction
Pipelines emerged to be the most efficient, economic and safe way of transporting fluids between two points. Municipal water distribution systems, oil and gas product delivery, all rely on the pipeline services (Reynold and Loren, 2000 and Henry, 2003). Pipe bends are usually characterized with unbalanced thrust forces that are usually balanced by thrust block or Joint restraint joints or combination of both (Dipra, 2006). Thrust block is usually defined as a mass concrete block design to provide resistance by transferring the thrust force to the soil through a larger surface area of the block such that the resultant stress does not exceed the lateral bearing strength of the soil (Dipra, 2006). Thrust block design is based on simple statics and does not depend on the type of pipe wall material (Jeyapalan and Rajah, 2007). The main aim of the design is to provide a bearing Area of the Block (Ab) that will distribute the force against the soil such that the lateral bearing strength of the soil was not exceeded. Figures, 1 and 2 presented the 2-Dimensional and 3-Dimensional View of the thrust block and bearing Area (Ab) clearly shown. In traditional approach the minimum bearing Area is calculated by solving the design equation but the problem is how to break the area in to the corresponding bearing width and height (b and h) which are the design variables. There are an infinitely many sets of values of bearing width and height (b and h) that will give that same bearing Area (Ab) but each set of b and h for the same Area (Ab) will yield different volume as the volume of the block was expressed as a function of bearing width and height. The aforementioned problem necessitated a need to come up with a tool that can search for the optimum values of the design variables for the bearing area (Ab) that would give minimum volume of the concrete. This paper therefore implores the characteristic traits of GAs in selecting the optimum values of b and h in thrust block design. Thrust blocks are not required on steel pipe with welded joints and thrust blocks are not required on steel or ductile-iron pipe with flanged joints if sufficient thrust restraint has been achieved by the restraint system (Water Agencies’ Standards, 2007). Washington Suburban Sanitary Commission (WSSC) (2008) puts it as part of the General Requirement that mass (unreinforced) concrete should be used for thrust block. Design of tie was not included herein as the joint movement is in the direction of the block and the block is purely in compression. WAS (2007) recommended tie rod only for Anchor Blocks located at unrestrained descending vertical bends but not for thrust blocks restraining horizontal bends.
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Figure 1 A 2–Dimensional View of Thrust Block (Chula Vista Fire Department)
Figure 2 A 3-Dimensional View of Thrust Block Genetic Algorithms (GAs) are search algorithms based on the mechanics of the natural selection process developed by John Holland (1975) over the course of the 1960s and 1970s and finally popularized by one of his students, David Goldberg, who was able to solve a difficult problem involving the control of gas pipeline transmission for his dissertation (Goldberg, 1989). This process mimics a natural population of biological creatures where successive generations of creatures are conceived, born, and raised until they are ready to reproduce (Chau et al., 2002). The most basic concept is that the strong tends to adapt and survive while the weak tends to die out. That is, optimization is based on evolution, and the "Survival of the fittest" concept. GAs has the ability to create an initial population of feasible solutions, and then recombine them in a way to guide their search to only the most promising areas of the state space. Each feasible solution is encoded as a chromosome (string) also called a genotype, and each chromosome is given a measure of fitness via a fitness (evaluation or objective) function. The fitness of a chromosome determines its ability to survive and produce offspring. A finite population of chromosomes is maintained. GAs use probabilistic rules to evolve a population from one generation to the next. The generations of the new solutions are developed by genetic recombination operators (Asifullah et al., 2008). It is very essential to design an efficient, versatile, reliable, unique and cost effective system based on optimum design approach and use of strong search methods as in Genetic Algorithm, Particle Swamp Optimization, Ant Colony Optimization, Taboo Search e.t.c. are necessary to obtain a reliability based optimal design (Bethany and Platt 2002, Hall 2005, Stehn and Johanssen 2002, Adedeji 2007, Adedeji 2010) 1.1.
Traditional design method of thrust block
The main aim of the design is to provide a bearing Area (Ab) of the Block that will distribute the force against the soil such that the lateral bearing strength of the soil was not exceeded. Equilibrium condition is shown in equation (1). 2
≤
44
ℎ
(1)
Epistemics in Science, Engineering and Technology, Vol. 2, No. 1, 2012, 43-50
Anwar et al
The required bearing Area of the thrust block is given in equation (2) below.
ℎ≥
(2)
The minimum bearing Area is given in equation (3):
ℎ=
(3)
The following are general criteria for bearing block design. •
• •
Bearing surface should, where possible, be placed against undisturbed soil. Where it is not possible, the fill between the bearing surface and undisturbed soil must be compacted to at least 90% Standard Proctor density. Block height (h) should be equal to or less than one-half the total depth to the bottom of the block, (Ht), but not less than the pipe diameter (D). Block height (h) should be chosen such that the calculated block width (b) varies between one and two times the height. =
(4)
where b= bearing width, h= bearing height, D= Diameter of the pipe, P = Internal Pressure of the liquid, θ= angle of the bend, Sf = Factor of safety, Sb = bearing strength of the soil (Dipra, 2006) 2.
Optimal Design Model
The problem was modeled using optimum design approach by generating the cost function and the corresponding constraints. The volume of the block was the function minimized. The volume of the block was arrived at by treating it as a truncated pyramid with bearing area of b x h and top area of D x Db/h angles of 45o and tan as shown in Fig. 2. The expression Db/h and tan were derived using parallel line property and similar triangles. All the dimensions of the block were calculated in terms of the design variables (b and h), so is the volume. Where b= bearing width, h= bearing height, D= Diameter of the pipe. The constraints include:
3.
Thrust – soil equilibrium constraint represents the basic design equation which guides the determination of the design variables such that the stress to be exerted against the soil by the block will not exceed the lateral bearing strength of the soil. In the traditional approach the design depends solely on the stability equation. Thrust - block equilibrium constraint ensures that the thrust stress being exerted against the block will not exceed the characteristic strength of the concrete. This prevents the thrust from crushing the block. The empirical design method does not consider this aspect. Non negativity constraints are constraints which make sure that the design variables are positive real numbers. Genetic Algorithm unless constrained search an optimum solution from -∞ to +∞, which makes it possible to arrive at negative values. Space constraints (Upper and lower bounds) involve constraining the design variables to fall within specified values. The reason for the above constraint in this project is that genetic Algorithm can find a global solution by making one of the variables extremely small (close to zero) and the other one very big which make the design unprofessional and unpractical. The design variables are then constrained to make one of the variables falls between one and two times the other. Model Application
MATLAB 7 G.A Toolbox was used to simulate the model. In this project double vector encoding was used to represent the population type. The Population size of 200 was used to represent the design variables. Rank Scaling was used as scaling function. Rank scales the raw scores based on the rank of each individual, rather than its score. The rank of an individual is its position in the sorted scores. The rank of the fittest individual is 1,
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Epistemics in Science, Engineering and Technology, Vol. 2, No. 1, 2012, 43-50
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the next fittest is 2, and so on. Rank fitness scaling removes the effect of the spread of the raw scores (MATLAB 7 G.A Toolbox). The method used for selecting the individuals from the mating pool that would be parents for the next generation was stochastic uniform method. Stochastic uniform lays out a line in which each parent corresponds to a section of the line of length proportional to its expectation. The algorithm moves along the line in steps of equal size, one step for each parent. At each step, the algorithm allocates a parent from the section it lands on. The first step is a uniform random number less than the step size (MATLAB 7 G.A Toolbox, 2007). Scattered crossover was used in the sense that, a random binary vector was created and the genes where the vector was a 1 from the father and the genes where the vector was 0 from the mother were selected, they were then genetically combined to produce the child. 3.1. Data The data used for simulation is as follows: P= 17000 KN/m2, D= 1m, θ= 30o (from Patterson) fs =24 N/mm2 and the bearing strength of the soils are presented in Table 1. P = Internal Pressure of the liquid, D = Diameter of the Pipe, θ= angle of the bend, fs =characteristic strength of the concrete Table 1 - Soil Horizontal Bearing Strength (Dipra 2006)
3.2.
Bearing Strength (KN/m2)
S/No.
Soil Type
1 2
Soft Clay Silt
47.88 71.82
3
Sandy Silt
143.64
4
Sand
191.52
5
Sandy Clay
287.28
6
Hard Clay
430.92
Problem formulation
The Objective function: Function to be minimized =
1 b h−D 6
b h
( )
where b= bearing width, h= bearing height, D= Diameter of the pipe Subject to: 2
sin ( ) ℎ≤0 2 (Thrust force – Soil Equilibrium constraint)
4
(6)
where P = Internal Pressure of the liquid, θ= angle of the bend, Sf = Factor of safety, Sb = bearing strength of the soil 2
sin
(
))
≤0 2 ℎ (Thrust force – Block Equilibrium constraint)
4
−
(7)
where fs =characteristic strength of the concrete. h, b > 0
(8) (Non-negativity constraint)
≤ℎ≤2
(5) (Space Constraint)
4.
Result
The following results shown in Figures 1 and 2 are graph of best and mean fitness and the Plot of the optimum design variables for soft clay respectively. The results for the remaining soils were not shown here.
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Epistemics in Science, Engineering and Technology, Vol. 2, No. 1, 2012, 43-50
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Anwar et al
Best and Mean fitness for Soft Clay
x 10
Best fitness Mean fitness 4
Fitness value
3
2
1
0
-1
0
5
10
15 Generation
20
25
30
Figure 3 Graph of best and mean fitness for soft clay Current Best Individual for Soft Clay 5 4.5
Current best individual
4 3.5 3 2.5 2 1.5 1 0.5 0
1
2 Number of variables (2)
Figure 4 Plot of the optimum design variables for soft clay Table 2 shows the values of bearing Area (Ab), bearing width and height obtained from genetic algorithm. Table 2 Results of basic parameters obtained from genetic algorithm S/No.
4.1.
Soil Type
Bearing Area (Ab) (m2)
Bearing width(b) (m)
Bearing height (h) (m)
1
Soft Clay
21.66
2
Silt
14.44
3.29062 2.68684
6.58123 5.37368
3
Sandy Silt
7.22
1.90000
3.80001
4
Sand
5.41
1.6455
3.29059
5
Sandy Clay
3.61
1.34417
2.68761
6
Hard Clay
2.41
2.19565
1.09738
Validation of the Model
For the genetic algorithm based design to be correct the product of the bearing width and height must the equal to the bearing Area calculated from the design equation. From Ductile Iron Pipe Research Association approach the bearing area is calculated as: = ℎ=
2 (
4
)sin ( ) 2
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(6)
Epistemics in Science, Engineering and Technology, Vol. 2, No. 1, 2012, 43-50
Anwar et al
Comparative bearing Areas calculated from the design equation and the genetic algorithm was presented in Table 3. Table 3 Areas generated by traditional method (design equation) and genetic algorithm Area(bh) m2 traditional approach
Soil
Area(bh) m2 genetic algorithm
Soft Clay
21.66
21.65
Silt
14.44
14.43
Sandy Silt
7.22
7.22
Sand
5.41
5.41
Sandy Clay
3.61
3.61
Hard Clay
2.41
2.41
4.2. Statistical Analysis Applying the most commonly means of evaluating error, known as Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) which are can be defined mathematically as follows: = =
∑ (
∑ |
)
(7)
|
where = manually calculated Area using traditional approach, Area, N = Number of samples
(8) = Genetic Algorithm based
Table 4 and 5 presented calculation for Root Mean Square Error and Mean Square Error. Table 4 Determination of root mean square error Soil
ATi - AGi
(ATi - AGi)2
ATi
AGi
Soft Clay
21.66
21.65
0.01
0.0001
Silt
14.44
14.43
0.01
0.0001
Sandy Silt
7.22
7.22
0.00
0.0000
Sand
5.41
5.41
0.00
0.0000
Sandy Clay
3.61
3.61
0.00
0.0000
Hard Clay
2.41
2.41
0.00
0.0000
Total
0.0002
Table 5 Determination of mean square error ATi - AGi
│ATi - AGi│
Soil
ATi
AGi
Soft Clay
21.66
21.65
0.01
0.01
Silt
14.44
14.43
0.01
0.01
Sandy Silt
7.22
7.22
0.00
0.00
Sand
5.41
5.41
0.00
0.00
Sandy Clay
3.61
3.61
0.00
0.00
Total
0.02
(A − A ) = 0.0002, N = 6 0.0002 = 0.0058 6 ∑ |A − A | MAE = N RMSE =
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|A − A | = 0.02 N=6 MAE =
0.02 = 0.058 6
Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) values are found to be 0.0058 and 0.058 respectively. This is the real indication that the model is accurate. The slight variation may be due to approximations. Genetic Algorithm results were compared with the five different randomly selected iterate values of the design variables for the same calculated bearing Area. It was found that the genetic algorithm based result provided the minimum volume of the concrete block. The volume of the block provided by the random iterates were higher than that of genetic algorithm by up to 95%, 90.97%, 74.55%, 59.72%, 33.77% and 119.05% for soft clay, silt, sandy silt, sand, sandy clay and hard clay respectively in contrast to Genetic Algorithm result. Table 6 shows the comparison between the volume of the block calculated from the five iterates and that of genetic algorithm for soft clay soil. And table 7 presented the percentage of volume increment of the iterates in relation to that of genetic algorithm for the same soft clay soils. Comparisons for the remaining soils were not presented here. Table 6 Comparison between iterated randomly selected dimensions and genetic algorithm based dimensions (soft clay) Method
Width (b) m
Height (h) m
Volume (V) m3
Area (A) m2
G.A
3.2906
6.5812
21.66
Iterate 1
3.5000
6.1886
21.66
Iterate 2
6.5813
3.2902
21.66
23.09
Iterate 3
4.3758
4.9500
21.66
15.67
Iterate 4
4.6540
4.6540
21.66
16.63
Iterate 5
5.0000
4.3320
21.66
17.83
11.84 12.58
G.A = Genetic Algorithm Table 7 Percentage of volume increase with respect to g.a (soft clay) Method
Iterate 1
= − (m3)
%
0.74
=
× 100
(%)
6.25
Iterate 2
11.25
95.00
Iterate 3
3.83
32.35
Iterate 4
4.79
40.46
Iterate 5
5.99
50.59
Iterate 1
0.74
6.25
dV = Volume change with respect to Genetic Algorithm %dV= Percentage of Volume change with respect to Genetic Algorithm 5.
Conclusion
In this study genetic algorithm was used to carry out an optimal design of bearing thrust block for horizontally bent ductile iron pipes. The following conclusions were drawn: i) The problem was modeled mathematically using optimization approach by generating an objective function and constraints. ii) Double vector population type was used to encode the population. Population size of 20 was used to represent the population size of each generation. Rank scale was employed as a scaling function and stochastic uniform as selection function while scattered cross over was used as the cross over function. iii) Results generated were successfully compared with the traditional approach results for validation. The values of the design variables given by genetic algorithm happened to be the optimum values which could hardly be arrived at using manual iteration. iv) Genetic Algorithm proved to be an efficient tool capable of being used for optimal design of bearing thrust block.
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References Adedeji, A.A. 2007. Genetic (evolutionary) algorithm: introduction and its use as an engineering design tool. 1st edition, Olad Publishers and Printing Enterprises. Adedeji, A.A. 2010. “Forget it”: A genetic concept and application in engineering design. Journal of Research Information in Civil Engineering, 7(2), pp 10-13. AsifUllah, K., T.K. Bandophadyaya, and S. Sudhir. 2008. Genetic algorithm based back propagation neural networks perform better than Backpropagation Neural Network in Stock Rates Prediction. International Journal of Computer Science and Network Security, 8 (7), pp 162-166. Bethany, S. and E.I. Platt. 2006. Parametric optimization of steel floor system using evolution. Published MSc Thesis, Department of Civil Engineering, Florida State University. Chau, K.W. and F. Albermani. 2002. Genetic algorithm for design of liquid retaining structure. Lecture Notes in Artificial Intelligence, 2358 pp 119-128. Ductile Iron Pipe Research Association (Dipra). 2006. Thrust restraint design for ductile iron pipe. 6th edition, Ductile iron pipe research association. Goldberg, D. E. 1989. Canonical Genetic Algorithm by Hand from Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA. Hall, S. 2005. Genetic Algorithm based optimized moment steel frames, Published MSc Thesis, Department of Civil Engineering, North Carolina State University. Henry, L. 2003. Pipeline Engineering. 1st edition, CRC Press. Holland J. H., 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press. Jeyapalan, J.K. and Rajah, S.K. 2007. Unified Approach to Thrust Restraint Design. Journal of Transportation Engineering ASCE 1(57), pp 57-61 Matlab 2007. Genetic Algorithm Toolbox Quick Reference. Reynold K. W. and R. A. Loren 2000. Structural Mechanics of Buried Pipes. 1st edition, CRC Press. Water Agencies’ Standards (WAS). 2007. Thrust Restraint and Anchor Block: Design Guidelines for Water and Sewer Facilities. Section 5.2., [www.google.com] Washington Suburban Sanitary Commission (WSSC). 2008. Thrust Restraint Design for Buried Pipelines: Common Design Guidelines. Part three. Section 27, C-27.1- C-27.26. Available from: www.wsscwater.com/communications/pipelinedesignmanual/part3/c-27-2008.pdf [www.google.com].
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