97

CHAPTER 5

DESIGN OF A RETRACTABLE ROOF STRUCTURE

5.1 General

The objective of this chapter is to perform the optimal design of a retractable roof structure

using the validated genetic algorithm. A predefined configuration of the movable panels

and the supporting structure is generated using a C-program developed as part of the

present research work. The structural design of retractable roof is carried-out against load

combination due to dead loads, live loads and wind loads. While carrying-out analysis

against wind, the permeability of roof structure is considered with roof full open, half open

and full closed conditions. Limit state method as per IS: 800-2007 is adopted for structural

steel design. The objective function minimizes the weight subjected to the constraints

imposed by strength and serviceability requirements. The problem formulation and the

step-by-step procedure for optimal design using real coded genetic algorithm with hybrid

mutation operator is explained in chapter 3.

5.2 Design of Movable Panels

The mechanism of movable roof system is attained by the sequence as detailed under. As

shown in Figure 3.4, panel B, panel C, panel E and panel F are the movable panels of the

retractable roof structure. The panels C and B move on the main arches whereas F and E

move on the quarter arches.

5.2.1 Panel C

The configuration of the retractable roof movable panel C generated by using a program

written in ‘C’ programming language is shown in Figure 5.1.

98

Problem statement of retractable roof movable panel C is as follows:

Number of members: 365, Number of joints: 172

The material properties: Elastic modulus, E= 2x105MPa, ϒ=2767.99 kg/m

3.

Allowable displacement: 50 mm. No. of member groups for each movable panel – 8 nos.

The design is as per IS: 800-2007.

The discrete design variable matrix - (the cross-sectional areas of members in sq. mm) as

follows:

[121 155 181 184 202 241 258 311 331 377 400 414 460 488 523 563 582 650

744 789 831 874 1010 1080 1120 1240 1270 1280 1470 1550 1730 1850 1860

1910 2050 2060 2090 2250 2270 2280 2320 2440 2490 2710 2760 2870 3190

3210 3480 3760 3950 4420 4950 6340 8650]

Loads on the movable panel C:

1. Dead load: Self weight of the members.

2. Live load: Joint load of 10 KN per each joint of movable panel.

3. Wind load: A horizontal point load of 20KN and a vertical point load of 15KN are

applied at each joint of the movable panel.

Figure 5.1 configuration of movable panel C

X Y Z

99

Optimal Design of panel C:

Optimal design of movable panel C is carried out using a population size of 20, 30 and 40

respectively. The results showing the optimal design for these population sizes are listed in

the Table 5.1 along with the corresponding design variables.

From the Table 5.1, the optimal weight of panel C for population size of 20, 30 and 40 is

739.16KN, 713.93KN and 655.77KN respectively. For calculating the loads on retractable

roof supporting structure the most optimum weight of 655.77KN is considered.

Table 5.1 Results of movable panel C

Design Variables

Population

Size(20)

(mm2)

Population

Size(30)

(mm2)

Population

Size(40)

(mm2)

A1 331.0 331.0 331.0

A2 331.0 331.0 331.0

A3 744.0 582.0 744.0

A4 331.0 331.0 331.0

A5 874.0 874.0 331.0

A6 582.0 582.0 582.0

A7 582.0 582.0 582.0

A8 241.0 241.0 241.0

Optimum Weight(KN): 739.16 713.93 655.77

The tabulated results are arrived at by using the two stage design procedure explained in

the section 3.9 of chapter 3. In the first stage the variation between the % of weak

individuals and weight of the truss is plotted by keeping the crossover probability constant

at 0.45. The second stage variation between the probabilities of crossover and weight of the

truss is plotted by keeping the % of weak individuals obtained from the first stage. Graphs

showing the variation of fitness with change in % of weak individuals and with change in

probability of crossover are presented in Figure 5.2 through Figure 5.4.

100

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 20 gives that 20% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.5 as shown in Figure 5.2.

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.2: Movable panel C design problem with population size – 20

101

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 30 gives that 20% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.55 as shown in Figure 5.3.

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.3: Movable panel C design problem with population size – 30

102

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 40 gives that 20% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.6 as shown in Figure 5.4.

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.4: Movable panel C design problem with population size – 40

103

5.2.2 Panel B

The configuration of the retractable roof movable panel B generated by using a program

written in ‘C’ programming language is shown in Figure 5.5.

Problem statement of retractable roof movable panel B is as follows:

Number of members: 283

Number of joints: 134

The material properties: Elastic modulus, E= 2x105MPa, ϒ=2767.99 kg/m

3.

Allowable displacement: 50 mm. No. of member groups for each movable panel – 8 nos.

The design is as per IS: 800-2007.

The discrete design variable matrix - (the cross-sectional areas of members in sq. mm) as

follows:

[121 155 181 184 202 241 258 311 331 377 400 414 460 488 523 563 582 650

744 789 831 874 1010 1080 1120 1240 1270 1280 1470 1550 1730 1850 1860

1910 2050 2060 2090 2250 2270 2280 2320 2440 2490 2710 2760 2870 3190

3210 3480 3760 3950 4420 4950 6340 8650]

Loads on the movable panel B:

1. Dead load: Self weight of the members.

2. Live load: Joint load of 10 KN per each joint of movable panel.

3. Wind load: A horizontal point load of 20KN and a vertical point load of 15KN are

applied at each joint of the movable panel.

104

Figure 5.5 configuration of movable panel B

Optimal Design of Panel B:

Optimal design of movable panel B is carried out using a population size of 20, 30 and 40

respectively. The results for these population sizes are listed in the Table 5.2 along with the

corresponding design variables. The results of the Table 5.2 are arrived at by using the two

stage design procedure explained in the section 3.9 of chapter 3. From the Table 5.2 the

optimum weight of panel B for population size of 20, 30 and 40 is 659.87KN, 650.25KN

and 638.87KN respectively. For calculating the loads on retractable roof supporting

structure the most optimum weight of 638.39KN is considered.

Table 5.2 Results of movable panel B

Design Variables

Population

Size(20)

(mm2)

Population

Size(30)

(mm2)

Population

Size(40)

(mm2)

A1 523.0 523.0 523.0

A2 400.0 400.0 400.0

A3 202.0 202.0 202.0

A4 202.0 202.0 202.0

A5 311.0 311.0 311.0

A6 2270.0 2270.0 2270.0

A7 414.0 311.0 184.0

A8 414.0 414.0 414.0

Optimum Weight(KN): 659.87 650.25 638.39

X Y Z

105

Graphs showing the variation of fitness with change in % of weak individuals and with

change in probability of crossover are presented in Figure5.6 through Figure 5.8.

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.6: Movable panel B design problem with population size – 20

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 20 gives that 20% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.55 as shown in Figure 5.6.

106

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.7: Movable panel B design problem with population size – 30

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 30 gives that 20% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.55 as shown in Figure 5.7.

107

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.8: Movable panel B design problem with population size – 40

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 40 gives that 15% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.6 as shown in Figure 5.8.

108

5.2.3 Panel F

The configuration of the retractable roof movable panel C generated by using a program

written in ‘C’ programming language is shown in Figure 5.9.

Problem statement of retractable roof movable panel F is as follows:

Number of members: 283

Number of joints: 134

The material properties: Elastic modulus, E= 2x105MPa, ϒ=2767.99 kg/m

3.

Allowable displacement: 50 mm. No. of member groups for each movable panel – 8 nos.

The design is as per IS: 800-2007.

The discrete design variable matrix - (the cross-sectional areas of members in sq. mm) as

follows:

[121 155 181 184 202 241 258 311 331 377 400 414 460 488 523 563 582 650

744 789 831 874 1010 1080 1120 1240 1270 1280 1470 1550 1730 1850 1860

1910 2050 2060 2090 2250 2270 2280 2320 2440 2490 2710 2760 2870 3190

3210 3480 3760 3950 4420 4950 6340 8650]

Loads on the movable panel F:

1. Dead load: Self weight of the members.

2. Live load: Joint load of 10 KN per each joint of movable panel.

3. Wind load: A horizontal point load of 20KN and a vertical point load of 15KN are

applied at each joint of the movable panel.

109

Figure 5.9 configuration of movable panel F

Optimal Design of Panel F:

Optimal design of movable panel F is carried out using a population size of 20, 30 and 40

respectively. The results for these population sizes are listed in the Table 5.3 along with the

corresponding design variables. The results of the Table 5.3 are arrived at by using the two

stage design procedure explained in the section 3.9 of chapter 3.

The optimum weight of panel F for population size of 20, 30 and 40 are 718.78KN,

715.20KN and 712.79KN respectively. For calculating the loads on retractable roof

supporting structure the most optimum weight of 712.79 is considered.

Table 5.3 Results of movable panel F

Design Variables

Population

Size(20)

(mm2)

Population

Size(30)

(mm2)

Population

Size(40)

(mm2)

A1 184.0 184.0 184.0

A2 311.0 377.0 377.0

A3 1240.0 1240.0 1240.0

A4 311.0 311.0 311.0

A5 331.0 331.0 155.0

A6 488.0 488.0 488.0

A7 1270.0 1270.0 1270.0

A8 400.0 331.0 400.0

Optimum Weight(KN): 718.78 715.20 712.79

X Y Z

110

Graphs showing the variation of fitness with change in % of weak individuals and with

change in probability of crossover are presented in Figure 5.10 through Figure 5.12.

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.10: Movable panel F design problem with population size – 20

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 20 gives that 20% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.5 as shown in Figure 5.10.

111

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.11: Movable panel F design problem with population size – 30

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 30 gives that 15% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.55 as shown in Figure 5.11.

112

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.12: Movable panel F design problem with population size – 40

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 40 gives that 20% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.55 as shown in Figure 5.12.

113

5.2.4 Panel E

The configuration of the retractable roof movable panel C generated by using a program

written in ‘C’ programming language is shown in Figure 5.13. Problem statement of

retractable roof movable panel E is as follows:

Number of members: 283, Number of joints: 134

The material properties: Elastic modulus, E= 2x105MPa, ϒ=2767.99 kg/m

3.

Allowable displacement: 50 mm. No. of member groups for each movable panel – 8 nos.

The design is as per IS: 800-2007.

The discrete design variable matrix - (the cross-sectional areas of members in sq. mm) as

follows:

[121 155 181 184 202 241 258 311 331 377 400 414 460 488 523 563 582 650

744 789 831 874 1010 1080 1120 1240 1270 1280 1470 1550 1730 1850 1860

1910 2050 2060 2090 2250 2270 2280 2320 2440 2490 2710 2760 2870 3190

3210 3480 3760 3950 4420 4950 6340 8650]

Loads on the movable panel E:

1. Dead load: Self weight of the members.

2. Live load: Joint load of 10 KN per each joint of movable panel.

3. Wind load: A horizontal point load of 20KN and a vertical point load of 15KN are

applied at each joint of the movable panel.

114

Figure 5.13 configuration of movable panel E

Optimal Design of Panel E:

Optimal design of movable panel E is carried out using a population size of 20, 30 and 40

respectively. The results for these population sizes are listed in the Table 5.4 along with the

corresponding design variables. The results of the Table 5.4 are arrived at by using the two

stage design procedure explained in the section 3.9 of chapter 3. The optimum weight of

panel E for population size of 20, 30 and 40 are 699.31KN, 647.84KN and 636.51KN

respectively. For calculating the loads on retractable roof supporting structure the most

optimum weight of 636.51KN is considered.

Table 5.4 Results of movable panel E

Design Variables

Population

Size(20)

(mm2)

Population

Size(30)

(mm2)

Population

Size(40)

(mm2)

A1 184.0 523.0 184.0

A2 1240.0 650.0 1240.0

A3 582.0 414.0 241.0

A4 155.0 155.0 155.0

A5 377.0 377.0 377.0

A6 488.0 563.0 488.0

A7 1270.0 1270.0 1270.0

A8 400.0 400.0 400.0

Optimum Weight(KN): 699.31 647.84 636.51

X Y Z

115

Graphs showing the variation of fitness with change in % of weak individuals and with

change in probability of crossover are presented in Figure5.14 through Figure 5.16.

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.14: Movable panel E design problem with population size – 20

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 20 gives that 15% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.5 as shown in Figure 5.14.

116

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.15: Movable panel E design problem with population size – 30

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 30 gives that 15% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.6 as shown in Figure 5.15.

117

Figure a: Variation of fitness with change in % of weak individuals

Figure b: Variation of fitness with change in probability of crossover

Figure 5.16: Movable panel E design problem with population size – 40

Execution of the genetic algorithm, with varied % of weak individuals for population size

of 40 gives that 15% of weak individuals can be allowed to get optimum fitness at

crossover probability of 0.65 as shown in Figure 5.16.

118

5.3 Design of supporting structure

The configuration of the retractable roof supporting structure generated by using a program

written in ‘C’ programming language is shown in Figure 5.17.

Problem statement of retractable roof structure is as follows:

Stadium diameter: 100 m., Height: 20 m., No. of members in retractable supporting

structure: 6628, No. of joints in retractable supporting structure: 1968, No. of degrees of

freedom of retractable supporting structure: 11448. The design is as per IS: 800-2007.

The material properties: Elastic modulus, E= 2x105MPa, ϒ=2767.99 kg/m

3.

Allowable displacement: 50 mm.

No. of member groups in the retractable roof supporting structure – 26 nos.

Figure: 5.17 Configuration of retractable roof supporting structure

The discrete design variable matrix - (the cross-sectional areas of members in sq. mm) as

follows:

[121 155 181 184 202 241 258 311 331 377 400 414 460 488 523 563 582 650

744 789 831 874 1010 1080 1120 1240 1270 1280 1470 1550 1730 1850 1860

1910 2050 2060 2090 2250 2270 2280 2320 2440 2490 2710 2760 2870 3190

3210 3480 3760 3950 4420 4950 6340 8650]

119

Loads on the supporting structure:

1. Dead load: Self weight of the members.

2. Live load: (a). Wheel loads from the movable panels on the respective joints.

Wheel load of Panel C: 281 KN (vertical) and 172KN (horizontal), Wheel load of

Panel B: 289 KN (vertical) and 167.5KN (horizontal), Wheel load of Panel F: 298.5

KN (vertical) and 167.5KN (horizontal), Wheel load of Panel E: 289 KN (vertical)

and 167.5KN (horizontal) (b). Joint load of 10 KN per joint at other joints.

3. Wind load: A horizontal point load of 20KN and a vertical point load of 15KN are

applied at each joint of upper surface.

Table 5.5: Optimal design of supporting structure

Design Variable Roof full open

(mm2)

Roof full closed

(mm2)

Roof half open

(mm2)

A1 831 460 460

A2 3950 2760 2760

A3 377 2050 874

A4 1240 181 181

A5 874 874 874

A6 258 582 582

A7 831 2090 2090

A8 3260 241 241

A9 2870 1850 3210

A10 311 1010 414

A11 1280 1850 1470

A12 3760 121 121

A13 789 3260 3260

A14 2250 4950 4950

A15 3480 414 414

A16 582 1850 1730

A17 2050 4420 181

A18 1910 202 202

A19 400 2060 2060

A20 2320 563 563

A21 2760 2870 2870

A22 789 650 650

A23 3950 1270 1270

A24 241 1550 1550

A25 563 2440 2440

A26 582 563 563

Weight(KN): 50937.78 57762.27 53299.88

120

Optimal design of supporting structure:

Optimal design of supporting structure for both movable and immovable panels and the

main frame work on which the panels move or rest are designed using the genetic

algorithm. Optimal design of the retractable roof supporting structure is carried out with

the roof in full open position, half open position and full closed position. The genetic

algorithm is executed according to the two stage process adopted ealier by considering

population size as forty for all the above cases of roof positions. In the first stage the

variation between the % of weak individuals and weight of the truss is plotted by keeping

the crossover probability constant at 0.45. The second stage variation between the

probabilities of crossover and weight of the truss is plotted by keeping the % of weak

individuals obtained from the first stage.

For the roof full open position of the supporting structure, execution of the genetic

algorithm with varied % of weak individuals for population size of forty gives that 15% of

weak individuals can be allowed to get optimum fitness at crossover probability of 0.45 as

shown in Figure 5.18.

Figure a: Variation of fitness with change in % of weak individuals

121

Figure b: Variation of fitness with change in probability of crossover

Figure 5.18: Retractable roof full open position with population size – 40

For the roof full closed position of the supporting structure, execution of the genetic

algorithm with varied % of weak individuals for population size of forty gives that 15% of

weak individuals can be allowed to get optimum fitness at crossover probability of 0.45 as

shown in Figure 5.19.

Figure a: Variation of fitness with change in % of weak individuals

122

Figure b: Variation of fitness with change in probability of crossover

Figure 5.19: Retractable roof full closed position with population size – 40

For the roof half open position of the supporting structure, execution of the genetic

algorithm with varied % of weak individuals for population size of 40 gives that 20% of

weak individuals can be allowed to get optimum fitness at crossover probability of 0.45 as

shown in Figure 5.20.

Figure a: Variation of fitness with change in % of weak individuals

123

Figure b: Variation of fitness with change in probability of crossover

Figure 5.20: Retractable roof half open position with population size – 40

5.4 Optimal design of retractable roof structure

Optimal design of the retractable roof movable panels and the supporting structure is

carried out using real coded genetic algorithm with hybrid mutation operator. Table: 5.1

through Table 5.4 shows the results of optimum design of movable panels. The optimum

weight of movable panels C and B which move on the main arches is 655.77 KN and

638.39 KN respectively. The optimum weight of movable panels F and E which move on

the quarter arches is 712.79 KN and 636.51 KN respectively. These optimal weights of

movable panels are obtained when the genetic algorithm is executed with a population size

of 40. Therefore, the optimal weight of retractable roof (movable panels) is 7985.52 KN.

Table: 5.5 shows the results of optimum design of retractable roof supporting structure

when the roof is full open, full closed and half open conditions. The optimum weight of

retractable roof supporting structure when the roof is in the above three positions is

50937.78 KN, 57762.27 KN and 53299.88 KN respectively.