# Optimimum Detailing Design Problem Of Reinforced Concrete Engineering Essay

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## CHAPTER 3

## 3.1 Introduction

Formulation of optimum design problem consists of identification of design variables, statement of objective function and constraints to be satisfied. In general, the structural optimization problem may be stated mathematically as,

Minimize

(3.1)

Subject to

(3.2)

where; f(x) is the objective function, gj (x )are the set of inequality constraints, x1, x2, x3,â€¦ xn, are design variables.

## 3.2 Objective Function

In structural optimization problems, the objective function is generally described as the weight or total cost of structure. Usually, the weight of structure is used for the optimum design of steel structures. For the optimum design of reinforced concrete (RC) structures, the cost of structure is more convenient as an objective function, because concrete structures involve different materials and the nature of concrete design is different. In reality, the minimum weight design may not be the minimum cost design for especially concrete structures. Besides the unit costs of different materials involved concrete construction influence the total cost of the concrete structures. For these reasons, the optimization problem of concrete structures should be formulated in terms of the total cost, which includes the costs of concrete, steel and formwork.

In this study, the objective function is selected as the total cost consisting of individual cost components of concrete, steel and formwork. The cost of any component is inclusive of material, fabrication, and labor. The objective function is expressed mathematically as,

Minimize

(3.3)

where, Cc = the cost of concrete, = the cost of reinforcing bars, = the cost of formwork (includes labor and placement) .

The costs of components, Cc, Cs, and Cf, are calculated for each member by the following equations.

For the cost of concrete, Cc;

(3.4)

where, Cc = the cost of concrete, Vc = the total volume of concrete, the unit cost of concrete ($/m3) .

(3.5)

where, Ncol = the number of column members, bi and di = the width and depth of ith column member with rectangular cross section, Lcolumn, i = the length of ith column member, Nbeam = the number of beam members, bw,j = the width of jth beam, hj = the height of jth beam, Lclear beam, j = the clear span length of jth beam.

For the cost of steel, Cs;

(3.6)

where, = the cost of steel, Ws = the total weight of steel used as reinforcement bar in the concrete frame, the unit cost of steel ($/kg) .

(3.7)

where, Nbar,.. = the number of longitudinal reinforcement bars placed in the member, Ntie,.. = the number of ties used in the member, As,.. = the area of reinforcement bars, lbar,.. = the length of reinforcement bars, Ast,.. , Ash,.. = the area of shear reinforcement bars (ties) ltie,.. = the length of ties.

For the cost of formwork, Cf ;

(3.8)

where, Cf = the cost of formwork, Af = the total formwork area, the unit cost of formwork ($/m2) .

(3.9)

where, Areabeams@joint,i = the cross-section area of beams connected to the ith column at joint.

The detailed evaluation of objective function is defined by Eqs. (3.3-3.9). The unit costs are based on market prices and their values changes from time to time and also from country to country. For these reasons, the unit price data cannot be fixed and needs to be updated. In the previous studies in the literature, researchers used different country design specifications for design and different market prices for unit costs depending to their countries. The different unit material costs are used for the design examples, and these values are given with concerned examples.

## 3.3 Design Variables

In this study, the column and beams in the structural frame are grouped to satisfy the applicability and efficiency of the introduced optimum design procedure. Also, design variables are divided two groups as columns design variables and beam design variables and to obtain practical designs beams and columns are separated to groups. Total number of design variables is determined according to number of column and beam groups.

## 3.3.1 Beam Design variables

For beam design groups, the cross-section of beams, the area of reinforcement bars along all beams, and the area of reinforcement bars placed on the top and bottom of beam spans and supports are considered as design variables. The cross-sectional dimensions of the beam are considered as the design variable. It is a general practice to provide the same width and height of beams in all spans. In addition to cross-section dimension of the beam, the areas of longitudinal bars that are placed continuously at the bottom of all the beams and the tensile reinforcements at the spans of beams, and supports for each beam are also considered as the design variable. These design variables and their numbers in the problem are defined in Table 3.1. The design variables relevant with reinforcement bars are not defined as the surface area of reinforcing bars. Instead of this, the reinforcement bar layout is defined and these design variables relevant with reinforcement are expressed in terms of the number of reinforcing bars and the diameter of reinforcing bars. By this way, design process can reach directly constructible optimum designs. In the literature, the similar approaches are used many researchers to obtain rational and practical designs [12,13,15,16,18,19].The details of the design variables are shown in Figure 3.1.

Table 3.1. Cross-sectional and reinforcement design variables for beam groups.

## Design Variables

## Number

Xi,1

The width and the height of beam cross-section

1

Xi,2

The area of the steel reinforcement that continues through the top of all the beams *

Ssasayisimi

1

Xi,3

The area of the steel reinforcement that continues through the bottom of all the beams *

1

Xi,4 -Xi,k

The area of the top steel reinforcement at spans of beams*

nbay

Xi, k+1 -Xi,m

The area of the bottom steel reinforcement at spans of beams*

nbay

Xi,m+1- Xi,n

The area of the top steel reinforcement at the supports*

nbay+1

Xi,n+1- Xi,Nbdv

The area of the bottom steel reinforcement at supports*

nbay+1

i= number of beam group (ith beam group), nbay ; number of spans (k = 3+ nbay , m= 3+2nbay, n= 4+3nbay, Nbdv= 5+4nbay )

* For the reinforcement design variables, the areas of steel reinforcement bars are defined in terms of the number and diameter of the bars (nï¦d n; number of bars, d; diameter of bars) to obtain constructable reinforcement areas.

The total number of the beam design variables, Nbdv, for one beam group changes according to number of span (bay) in frame. Total number of beam design variables for one beam group is calculated by Eqn.3.10 ;

(3.10)

The total number of beam design variables when each beam group considered, Ndvbeam , is computed by following equation;

(3.11)

where, Nbdv = the total number of the beam design variables for one beam group, nbay = the number of bays in a frame, nbeam group = the total number of beam group.

Figure 3.1. The reinforcement bar layout and the design variables for ith beam group.

Table 3.2. Variable poll for beam design variables

## #

## X1 (mm)

## #

## X2 - X3

## #

## X4 - XNbdv

1

250/400

1

2ï¦12

1

NNR

2

250/450

2

2ï¦14

2

1ï¦12

3

250/500

3

3ï¦12

3

1ï¦14

4

250/550

4

2ï¦16

4

1ï¦16

5

250/600

5

3ï¦14

5

1ï¦18

6

300/400

.6

3ï¦16

6

1ï¦20

7

300/450

7

1ï¦22

8

300/500

8

1ï¦24

9

300/550

9

1ï¦26

10

300/600

10

1ï¦28

11

300/650

11

1ï¦30

12

300/700

12

2ï¦12

13

350/500

13

2ï¦14

14

350/550

14

2ï¦16

15

350/600

15

2ï¦18

16

350/650

16

2ï¦20

17

350/700

17

2ï¦22

18

400/600

## .

## .

19

400/650

## .

## .

20

400/700

## .

## .

21

400/750

## .

## .

22

400/800

## .

## .

23

450/700

## .

## .

24

450/750

## .

## .

25

450/800

45

5ï¦18

26

450/850

46

5ï¦20

27

450/900

47

5ï¦22

28

500/700

48

5ï¦24

29

500/750

49

5ï¦26

30

500/800

50

5ï¦28

31

500/850

51

5ï¦30

32

500/900

52

6ï¦12

53

6ï¦14

54

6ï¦16

55

6ï¦18

56

6ï¦20

57

6ï¦22

58

6ï¦24

59

6ï¦26

60

6ï¦28

61

6ï¦30

* NNR ; no need reinforcement

For the cross-sectional and reinforcement design variables, the design variable pools are created. For the width of the beam the values in pool starts from 250 mm goes upto 500 mm with the increment of 50mm, similarly for the height of the beam the values starty from 400 mm increases upto 900 mm with the same increment and the combinations of width and height values are is composed in the variable pool table. For the reinforcement design variables, the values are selected such that they give constructable reinforcement areas. In other words, the number and diameter of the bar for a beam can be considered as one design variable (nï¦d n; number of bars, d; diameter of bars). For this purpose, the variable pool table is composed of a combination of number and diameter of bars. For the reinforcement design variables, the design variable pool is created by the combination of number and diameter of reinforcement bars. The number of reinforcement bars changes between 2 and 6 and the diameter of bars changes between 12 and 30 with an increment of 2 mm. Then 61 different combination of number and diameter of a bar are arranged for each reinforcement design variables. The 6 reinforcement bar combination are used for the reinforcement continues through the top (Xi,2) and bottom (Xi,3) of all the beams. The variable pool composed for reinforcement design variables is given in Table 3.2.

In this optimum design problem, material strengths, unit costs of materials, structural geometry, support conditions, loading conditions, and cover details are pre-assigned as design parameters at the beginning of the optimization process. However, the value of the dead load which includes the self-weight of beam depending on the cross-sectional dimensions is automatically updated during the design cycles.

## 3.3.2 Column Design Variables

The optimum design problem of RC buildings is more complex than the optimal design of steel structures because only one structural material is considered in steel structures and the cost of the steel structures is only assumed to be proportional to its weight. Unlike the design of steel structures, there is infinite set of member size and amount of steel reinforcement used in the design of reinforced concrete (RC) buildings. The importance difference between optimum designs of steel and concrete structures is that more combinatorial characteristics affect the determining the cross-sectional dimensions, and the layout and arrangements of reinforcing bars in RC building design. In addition to the discrete and combinatorial nature of the RC sections, the restrictions and reinforcement detailing specified in the design specifications make the optimum design of RC buildings even more complicated.

In the literature, many researchers suggested and used a lot of practical discrete optimization techniques developed for discrete optimum design of RC frames by the construction of a concrete section database [19,23,24,25,26,27]. In these studies, discrete optimum sections are directly searched based on the relationship between the section identification numbers and properties (dimensions and resistant capacities) of sections. The sections widely used in practical design are selected to construct database.

In this thesis, the concrete section database is constructed for the selection of section of column members. Practically, the dimensions of column sections in a RC frames are usually increased by 50 mm a step, and the diameters of reinforcing bars in column members change between 14 and 30 mm most frequently.

Table 3.3 Design variable bounds for RC column design examples.

Design Variables

Lower bound

Upper Bound

Increment

(Step size)

Number of possible values in the range

## X1c

b (mm)

300

500

50 mm

5

## X2c

d (mm)

400

1000

50 mm

13

## X3c

ï¦ï€ ï€ ï€ (mm)

{ 14,16,18,20,22,24,26,28,30}

9

X4c and X5c

n1 and n2

0

7

1

8

Figure 3.2. The design variables for construction of column section database.

In this study, the design variables for the construction of column section database are selected as the dimensions of columns in x and y directions, the diameter of reinforcement bars at the cross-section of column and numbers of reinforcement bars in both sides of the column as shown in Figure 3.2. The lower-upper bounds and increments of these variables are given in Table 3.3. Assigning these values to variables all possible section combinations are composed, and the combinations satisfy the constraints between 3.14 and 3.22 (given in section 3.3.1) are taken the section pool. However, the number of possible combinations is 37440 and obtained feasible sections number is obtained as 6199. This quantity of feasible column sections is too large to obtain the optimum results using optimization methods. Additionally, some of sections have similar section properties and flexural moment strength. For these reasons, obtained feasible column sections are evaluated to their section properties and flexural strength for balanced case. Finally, the feasible solution number in the column section database is decreased to 219 which is a reasonable number for optimum design process. The selected feasible column sections are given in Table 3.4. The compressive strength of concrete and the yielding strength of steel are taken 30MPa and 400MPa, respectively, and the cover of concrete section is taken 50mm to calculate the balanced flexural strength of feasible concrete column sections in the database (Table 3.4).

Table 3.4 Design variables database for column sections.

## #

## d

## b

## f

## n1

## n2

## (kN) max,nP

## (kN) bP

## (mkN) bM

1

400

300

14

1

1

2288,78

851

130,97

2

400

300

14

1

2

2363,72

848,46

142,13

3

400

350

14

2

1

2695,22

991,85

149,3

4

400

350

14

1

2

2695,22

991

158,38

5

400

350

22

1

1

3060,77

972,18

193,67

6

400

400

16

2

1

3141,44

1128,26

175,97

7

400

400

18

1

1

3147,55

1125,48

184,34

8

400

400

30

1

2

4372,67

1069,63

322,26

9

450

300

14

2

1

2612,35

978,21

165,86

10

450

300

18

3

1

2980,95

978,22

198,67

11

450

350

18

2

1

3230

1132,58

212,82

12

450

350

20

5

3

4140,05

1134,76

304,82

13

450

400

14

4

3

3658

1305,69

241,09

14

450

400

14

3

4

3658

1302,73

250,54

15

500

300

14

3

2

3010,86

1110,75

221,56

16

500

300

14

5

2

3160,75

1119,95

231,13

17

500

300

22

2

1

3411,59

1093,26

269,78

18

500

300

20

3

2

3556,89

1104,83

289,79

19

500

350

16

2

2

3487,95

1283,73

261,31

20

500

350

18

3

1

3643,95

1294,64

265,44

21

500

350

16

2

3

3585,83

1280,4

280,92

Table 3.4 Cont. Design variable database for column sections.

## #

## d

## b

## f

## n1

## n2

## (kN) max,nP

## (kN) bP

## (mkN) bM

22

500

400

20

2

1

4079,74

1462,73

301,26

23

500

400

26

3

1

4865,9

1466,68

380,36

24

500

400

24

4

2

5076,97

1464,93

410,59

25

500

450

18

2

3

4596,59

1641,79

358,32

26

500

450

18

2

4

4720,48

1637,58

383,01

27

500

450

24

2

2

5050,85

1627,56

410,79

28

550

300

18

2

1

3354,32

1229,6

275,49

29

550

300

18

3

1

3478,2

1242,8

284,23

30

550

300

20

2

1

3499,62

1227,39

295,86

31

550

300

20

4

1

3805,51

1248,66

316,54

32

550

300

20

2

2

3652,57

1222,19

330,19

33

550

350

14

5

2

3865,19

1456,02

305,28

34

550

350

14

3

3

3790,25

1441,67

311,33

35

550

350

20

2

1

3955,43

1431,02

326,84

36

550

350

18

2

2

3934,02

1429,43

334,43

37

550

350

20

6

2

4720,17

1468,34

404,48

38

550

400

14

6

2

4395,95

1664,26

341,66

39

550

400

18

3

2

4513,72

1646,67

374,16

40

550

400

16

2

4

4429,6

1630,45

385,69

41

550

400

24

2

1

4747,73

1627,79

404,4

42

550

400

20

5

3

5175,98

1661,33

459,15

43

550

450

16

2

3

4787,52

1838,24

394,51

44

550

500

16

5

2

5439,11

2071,55

424,56

45

550

500

14

2

6

5307,57

2040,23

450,22

46

600

300

14

4

1

3508,11

1374,17

294,71

47

600

300

18

3

2

3850,72

1368,77

362,88

48

600

350

16

3

1

4068,07

1596,43

346,04

49

600

350

16

4

1

4165,96

1601,92

353,42

50

600

350

18

3

1

4224,08

1597,8

368,63

51

600

350

16

6

3

4557,51

1612,9

419,59

52

600

350

16

7

3

4655,39

1625,05

428,66

Table 3.4 Cont. Design variables database for column sections.

## #

## d

## b

## f

## n1

## n2

## (kN) max,nP

## (kN) bP

## (mkN) bM

53

600

350

20

3

2

4551,39

1594,04

432,05

54

600

350

22

3

3

4961,29

1588,18

513,62

55

600

400

16

4

1

4663,21

1827,15

390,38

56

600

400

16

7

1

4956,87

1856,95

416,12

57

600

400

18

3

2

4845,22

1818,41

436,78

58

600

400

18

4

2

4969,11

1825,03

446,05

59

600

400

22

3

2

5273,47

1818,46

504,35

60

600

400

18

4

4

5216,88

1816,61

508,45

61

600

450

16

3

2

5160,46

2043,57

444,72

62

600

450

16

4

3

5356,23

2045,72

476,85

63

600

450

16

3

4

5356,23

2036,9

494,23

64

600

450

24

3

2

6016,97

2041,96

579,72

65

600

450

30

3

5

7916,59

2003,81

966,49

66

600

500

18

4

3

6087,49

2270,45

551,15

67

600

500

20

4

2

6196,09

2274,99

554,22

68

600

500

16

6

4

6147,14

2285,27

555,23

69

600

500

20

4

4

6501,98

2264,58

630,92

70

600

500

26

3

5

7557,33

2238,09

850,07

71

650

300

22

3

1

4342,53

1508,58

440,63

72

650

350

16

3

3

4553,91

1738,76

455,28

73

650

350

24

3

1

5092,29

1756,4

517,29

74

650

350

22

6

3

5806,56

1787,03

636

75

650

400

14

5

4

5133,89

1998,02

498,79

76

650

400

18

3

4

5424,49

1980,79

572,46

77

650

450

16

5

5

6022,83

2244,12

614,87

78

650

500

18

5

3

6625,76

2500,91

647,58

79

650

500

20

5

3

6916,36

2505,33

696,33

80

650

500

20

3

6

7069,31

2459,97

795,58

81

650

500

24

3

5

7589,33

2459,47

889,61

82

700

300

14

5

1

4080,31

1641,18

399,37

83

700

300

20

4

2

4704,34

1640,9

527,01

Table 3.4 Cont. Design variables database for column sections.

## #

## d

## b

## f

## n1

## n2

## (kN) max,nP

## (kN) bP

## (mkN) bM

84

700

300

20

5

2

4857,29

1662,59

543,7

85

700

300

26

3

1

5031,65

1649,01

576,21

86

700

300

24

3

2

5022,47

1636,51

603

87

700

350

14

4

2

4660,43

1894,71

464,48

88

700

350

14

3

3

4660,43

1886,1

480,4

89

700

350

16

3

2

4746,08

1890,53

488,73

90

700

350

18

6

1

5175,87

1933,86

523,74

91

700

350

16

7

2

5137,63

1934,52

529,61

92

700

350

18

7

1

5299,76

1951,72

537,78

93

700

350

16

7

3

5235,52

1931,19

559,51

94

700

350

24

3

1

5382,35

1909,13

587,06

95

700

350

20

5

2

5437,41

1928,55

594,11

96

700

350

20

4

3

5437,41

1901,65

623,8

97

700

350

28

3

1

5859,55

1918,42

669,01

98

700

400

16

3

4

5521,98

2150,65

598,97

99

700

400

18

5

4

6003,77

2178,17

675,01

100

700

400

24

3

4

6623,21

2151,78

836,42

101

700

450

16

4

3

6102,11

2428,44

628,66

102

700

450

16

3

5

6200

2414,11

679,3

103

700

450

20

5

2

6597,66

2460,47

694,93

104

700

450

24

4

2

6983,09

2447,14

774,25

105

700

450

22

5

4

7256,87

2456,04

856,16

106

700

500

16

6

3

6878,01

2717,23

699,65

107

700

500

18

6

2

7040,13

2728,77

712,71

108

700

500

20

4

6

7636,63

2683,91

914,2

109

700

500

28

3

5

8799,04

2670,53

1178,62

110

750

300

14

4

2

4328,93

1754,73

468,85

111

750

300

16

4

1

4414,58

1762,82

472,77

112

750

300

14

5

2

4403,88

1766,52

477,93

113

750

300

16

5

1

4512,47

1778

484,58

114

750

300

18

5

1

4720,48

1788,23

520,22

Table 3.4 Cont. Design variables database for column sections.

## #

## d

## b

## f

## n1

## n2

## (kN) max,uP

## (kN) bP

## (mkN) bM

## #

## d

## b

## f

## n1

## n2

## (kN) max,nP

## (kN) bP

## (mkN) bM

115

750

300

18

6

1

4844,37

1798,77

533,7

116

750

300

18

4

2

4720,48

1765,06

546,31

117

750

300

20

5

1

4952,96

1799,4

559,81

118

750

300

22

4

2

5209,92

1777,7

642,12

119

750

350

14

4

3

5025,44

2040,18

551,75

120

750

350

18

4

1

5218,15

2056,42

563,28

121

750

350

14

6

3

5175,33

2058,67

569,08

122

750

350

16

5

3

5329,81

2058,9

607,47

123

750

350

18

5

2

5465,93

2071,17

619,12

124

750

350

22

5

1

5831,48

2097,86

661,23

125

750

350

20

6

2

5880,42

2093,61

684,68

126

750

350

20

4

3

5727,47

2052,64

700,28

127

750

350

22

4

3

6016,55

2057,72

760,82

128

750

400

16

7

2

6049,26

2373,07

655,75

129

750

400

18

6

4

6459,16

2360,44

772,46

130

750

400

26

5

1

7040,36

2409,29

817,69

131

750

400

26

4

2

7040,36

2363,13

871,63

132

750

400

22

7

3

7193,31

2414,07

883,36

133

750

450

16

5

2

6475,05

2637,37

690,89

134

750

450

14

5

4

6418,45

2625,39

701,69

135

750

450

14

4

5

6418,45

2611,07

717,55

136

750

450

16

5

5

6768,71

2627,37

788,31

137

750

450

28

7

3

9191,41

2752,48

1193,5

138

750

500

20

4

5

7898,06

2902,44

974,86

139

800

350

14

6

2

5390,45

2209,98

611,59

140

800

350

14

5

3

5390,45

2200,18

629,53

141

800

350

20

4

2

5864,59

2208,09

725,95

142

800

350

20

5

2

6017,54

2232,38

745,8

143

800

350

18

7

3

6127,66

2246,81

767,84

144

800

350

20

7

2

6323,43

2268,88

783,91

145

800

350

26

4

1

6450,38

2240,46

812,38

Table 3.4 Cont. Design variables database for column sections.

## #

## d

## b

## f

## n1

## n2

## (kN) max,nP

## (kN) bP

## (mkN) bM

146

800

350

20

6

3

6323,43

2241,05

818,25

147

800

350

28

5

2

7339,01

2285,63

1010,52

148

800

400

14

7

2

6128,39

2529,5

687,36

149

800

400

18

4

1

6171,22

2513,61

698,3

150

800

400

20

4

1

6374,64

2520,81

737,48

151

800

400

18

4

4

6542,88

2500,98

830,96

152

800

400

20

5

4

6986,43

2529,48

920,61

153

800

400

24

5

2

7286,21

2564,06

933,75

154

800

450

18

5

2

7081,99

2837,24

824,64

155

800

450

20

6

2

7496,48

2861,28

895,73

156

800

450

24

4

5

8389,7

2814,16

1205,06

157

800

500

20

5

2

8006,54

3154,92

943,66

158

800

500

24

5

5

9272,95

3154,95

1299,31

159

850

350

14

7

1

5680,51

2371,26

664,1

160

850

350

16

7

1

5909,93

2386,99

707

161

850

350

14

7

3

5830,4

2366,19

721,87

162

850

350

16

6

2

5909,93

2369,95

731,11

163

850

350

18

4

2

5922,17

2350,24

752,5

164

850

350

22

4

2

6411,6

2366

864,17

165

850

350

22

5

2

6596,67

2396,32

889,99

166

850

350

20

6

3

6613,5

2392,78

905,04

167

850

350

24

4

2

6693,03

2374,99

927,95

168

850

350

24

5

2

6913,27

2410,66

958,55

169

850

350

28

5

1

7329,29

2452,34

999,13

170

850

350

24

6

3

7353,77

2423,86

1070,05

171

850

350

28

5

2

7629,07

2442,15

1112,4

172

850

400

16

6

2

6614,37

2699,07

805,64

173

850

400

20

5

1

6859,09

2716,56

843,15

174

850

400

18

7

2

6998,27

2729,54

877,44

175

850

400

22

4

2

7116,04

2693,88

938,66

Table 3.4 Cont. Design variables database for column sections.

## #

## d

## b

## f

## n1

## n2

## (kN) max,nP

## (kN) bP

## (mkN) bM

176

850

450

20

5

3

7869,42

3034,44

1034,54

177

850

450

20

6

3

8022,37

3049,36

1054,05

178

850

450

18

5

5

7826,6

3015,77

1061,38

179

850

450

22

5

5

8560,75

3033,17

1250,53

180

850

500

16

4

6

8219,02

3317,52

1078,72

181

850

500

30

4

2

9797,45

3383,91

1373,69

182

900

400

18

6

2

7205,88

2882,64

951,43

183

900

400

18

5

3

7205,88

2865,38

985,08

184

900

400

18

7

3

7453,66

2895,41

1020,17

185

900

400

22

6

2

7817,68

2914,96

1088,25

186

900

450

16

5

4

7789,63

3201,47

1043,27

187

900

450

20

5

3

8242,36

3226,06

1141,62

188

900

500

20

5

5

9294,13

3564,72

1350,11

189

950

400

20

5

1

7522,09

3057,72

1027,23

190

950

400

18

6

3

7661,27

3048,44

1100,64

191

950

400

20

5

3

7827,98

3047,31

1160,2

192

950

400

28

5

2

8996,51

3119,62

1420,16

193

950

400

26

6

3

9141,81

3119,23

1477,93

194

950

450

18

6

3

8448,58

3418,7

1193,8

195

950

450

20

7

2

8768,25

3459,56

1232,71

196

950

450

24

5

4

9508,52

3437,63

1527,68

197

950

450

26

6

3

9929,12

3487,83

1571,04

198

950

450

28

5

3

10083,6

3477,61

1642,28

199

950

500

20

7

2

9555,56

3829,41

1325,86

200

950

500

28

5

4

11170,7

3835,6

1864,41

201

950

500

30

5

4

11658,6

3851,41

1999,42

202

1000

400

20

5

1

7853,59

3227,6

1125,5

203

1000

400

22

6

1

8295,61

3264,47

1215,93

204

1000

400

22

5

2

8295,61

3238,12

1272,4

205

1000

400

24

5

4

9052,7

3240,31

1557,86

206

1000

400

26

6

3

9473,31

3293,3

1604,32

Table 3.4 Cont. Design variables database for column sections.

## #

## d

## b

## f

## n1

## n2

## (kN) max,nP

## (kN) bP

## (mkN) bM

207

1000

400

30

5

4

10415,5

3291,11

1960,43

208

1000

450

26

6

3

10302,1

3682,68

1707,54

209

1000

450

26

5

4

10302,1

3645,65

1785,85

210

1000

450

26

6

4

10560,5

3673,9

1825,85

211

1000

450

26

5

5

10560,5

3636,86

1904,17

212

1000

450

28

5

5

11056,1

3651,94

2056,94

213

1000

450

30

5

5

11588,4

3667,96

2220,41

214

1000

500

20

5

2

9664,04

4003,65

1402,51

215

1000

500

20

7

2

9969,93

4041,03

1450,92

216

1000

500

20

5

3

9816,98

3998,45

1473,01

217

1000

500

18

5

5

9774,16

3977,65

1504,87

218

1000

500

22

6

4

10508,3

4025,99

1677,74

219

1000

500

28

5

2

10985,5

4071,49

1749,45

## 3.4 Constraints

Constraints to be considered in the optimum design problem are strength, serviceability, ductility and other side constraints. Constraints can be imposed separately for column groups, beam groups and connection regions. These constraints are taken from ACI 318-05.

## 3.4.1 Constraints for Column Groups

Two types of design constraints are considered for the column members of the RC frame. The first type includes those constraints on the axial load and moment resistance capacities of the section, clear spacing limits between reinforcing bars, and the minimum and the maximum percentage of steel allowed. The second type consists of those constraints defining architectural requirements and good design and detailing practices. These include the requirement of the minimum and the maximum dimensions of column, the maximum aspect ratio of the section, maximum number of reinforcing bars and other reinforcement requirements. The constraints are explained and expressed in a normalized form as given below.

The maximum axial load capacity of columns, Pn,max should be greater than the factored axial design load acting on the column section, Pd ;

(3.12)

where, i = number of the column (ith column), Ncol = total number of columns, j = load combination type.

For a column section with uni-axial bending, the moment carrying capacity of column section, Mn, obtained for each factored axial design load, Pd, should be greater than the applied factored design moment, Md ;

(3.13)

where, i = number of the column (ith column), Ncol = total number of columns, j = load combination type.

The percentage of longitudinal reinforcement steel, Ï, in a column section should be between minimum and maximum limits permitted by design specification (Ïmin= 0.01 and Ïmax = 0.06) ;

(3.14)

and

(3.15)

where, i = number of the column (ith column), Ncol = total number of columns.

The width b and the height d of a column section should not be less than the minimum dimensions limit value given for columns (min. dimension, cdmin = 300mm);

(3.16)

and

(3.17)

where, i = number of the column (ith column), Ncol = total number of columns.

The ratio of shorter dimension of column section to longer one should be greater than permitted limit (cdrmin = 0.40);

(3.18)

where, i = number of the column (ith column), Ncol = total number of columns.

The minimum diameter of longitudinal reinforcing bars, Ø, in a column section should be greater than minimum bar diameter, Ømin, specified by design code;

(3.19)

where, i = number of the column (ith column), Ncol = total number of columns.

The total number of longitudinal reinforcing bars, nrb, in a column section should be smaller than specified maximum number of reinforcing bars, nrbmax, for detailing practice (nrbmax = 24);

(3.20)

where, i = number of the column (ith column), Ncol = total number of columns.

The minimum and maximum clear spacings between longitudinal bars, a, in a column section should be between minimum and maximum limits, amin and amax, specified for detailing practice (amin = 50 mm and amax = 150 mm);

(3.21)

and

(3.22)

where, i = number of the column (ith column), Ncol = total number of columns.

The shear force capacity of column section, ØVn, should be greater than applied factored design shear force, Vd ;

(3.23)

where, i = number of the column (ith column), Ncol = total number of columns, j = load combination type.

Also, the shear force capacity of column section, ØVn, should be greater than the minimum capacity shear forces, min{Vec , Veb }, based on probable maximum flexural strength of column, Vec , and based on probable maximum flexural strengths at the ends of beams framed into the top joint of column, Veb ;

(3.24)

where, k = number of the column group (kth column group), Ncol = total number of column groups.

The factored design shear force acting on column section, Vd, should be less than allowed maximum shear force capacity, ØVmax ;

(3.25)

The area of shear reinforcement (ties), Ash , should be greater than limitations on the minimum area of shear reinforcement, Ash,min ;

(3.26)

(3.27)

where, k = number of the column group (kth column group), Ncolumn group = total number of column groups, j = three (top, middle and bottom parts) shear design region of column , fc = the compressive strength of concrete, fy = the yielding strength of reinforcing steel, b and d = the width and height of column section, s = the spacing between stirrups (ties), bc = the cross-sectional dimension of column core measured center-to-center of outer legs of the transverse reinforcement comprising area Ash , Ag = the gross area of section, Ach = the cross-sectional area of member measured out-to-out of transverse reinforcement.

The spacing between stirrups, s, in the column should be greater than minimum spacing, smin (smin = 50 mm) for constructional requirements;

(3.28)

where, k = number of the column group (kth column group), Ncolumn group = total number of column groups, j = three (top, middle and bottom parts) shear design region of the column.

At the top and bottom ends of column members, the spacing between stirrups, s, should be less than maximum spacing of shear reinforcement for end regions of column, smax,end ;

(3.29)

(3.30)

where, k = number of the column group (kth column group), Ncolumn group = total number of column groups, j = two (top and middle parts) shear design region of the column, b and d = the width and height of column section, s = the spacing between stirrups (ties), Øb = the diameter of longitudinal reinforcing bars.

For the middle parts column members, the spacing between stirrups, s, should be less than maximum spacing of shear reinforcement, smax,middle ;

(3.31)

(3.32)

where, k = number of the column group (kth column group), Ncolumn group = total number of column groups, Øb = the diameter of longitudinal reinforcing bars.

The length of top and bottom shear regions, lo, should be greater than allowable design length, lo,min ;

(3.33)

(3.34)

where, k = number of the column group (kth column group), Ncolumn group = total number of column groups, j = two (top and middle parts) shear design region of column, b and d = the width and height of column section.

The detailed explanation of design philosophy of columns subject to ACI 318-05 specifications and the derivation of the design constraints accordingly are given in Appendix B.

## 3.4.2 Constraints for Beam Groups

Constraints to be imposed for each beam group are based on strength, serviceability, ductility and other side constraints. The normalized forms of all constraints considered in optimum design problem are given below.

For three critical sections (left end, middle part and right end) of each beam, the negative (with top steel in tension) and positive (with bottom steel in tension) moment carrying capacities of section for Mn, should be greater than the applied factored design moments Md ;

(3.35)

where, i = number of the beam (ith beam), Nbeam = total number of beams, j = load combination type, k = three critical sections for flexural design of beam, l= the negative moment and positive moment situations (top steel in tension or bottom steel in tension).

The tension area of longitudinal reinforcement steel bars in tension As, for three critical sections in a beam should satisfy the minimum and the maximum requirements permitted by design specification (Ïmin= 0.01 and Ïmax = 0.06) ;

(3.36)

and

(3.37)

(3.38)

where, i = number of the beam (ith beam), Nbeam = total number of beams, k = three critical sections for flexural design of beam, l= the negative moment and positive moment situations (checking for top and bottom reinforcement steel area), bw and h = the width and height of beam section, fc = the compressive strength of concrete, fy = the yielding strength of reinforcing steel.

At any end (support) of the beams, the positive moment capacity Mn- , (i.e., associated with the bottom steel) should be greater than 1/2 of the beam negative moment capacity Mn+, (i.e., associated with the top steel) at that end;

(3.39)

where, i = number of the beam (i.th beam), Nbeam = total number of beams, k = the left and right ends of beam.

The positive flexural moment strength, Mn,middle+ ,at span of beams should be greater than a quarter of negative and positive flexural moment strengths at the ends of beams, Mn,- and Mn+ ;

(3.40)

and

(3.41)

where, i = number of the beam (i.th beam), Nbeam = total number of beams, k = the left and right ends of beam.

The shear force capacity of three regions (left and right ends, and span) of beam, ØVn, should be greater than applied factored design shear force, Vd ;

(3.42)

where, i = number of the beam (i.th beam), Nbeam = total number of beams, j = load combination type, k = three critical region (left and right ends, and span) for shear design of beam.

Also, the shear force capacity of design regions in a beam, ØVn, should be greater than the probable shear forces based on probable maximum flexural strengths with the factored gravity loads at beam ends of beam, max{Ve1 , Ve2 } ;

(3.43)

where, where, i = number of the beam (i.th beam), Nbeam = total number of beams, k = three critical region (left and right ends, and span) for shear design of beam.

The factored design shear forces at the middle and ends of beam, Vd, should be less than allowed maximum shear force capacity, ØVmax ;

(3.44)

where, where, i = number of the beam (i.th beam), Nbeam = total number of beams, k = three critical region (left and right ends, and span) for shear design of beam.

The spacing between stirrups s, in the middle and the ends of the beam should be greater than the minimum spacing, smin (smin =50 mm) for constructional requirements;

(3.45)

where, where, i = number of the beam (i.th beam), Nbeam = total number of beams, k = three critical region (left and right ends, and span) for shear design of beam.

At the left and right ends of beam members, the spacing between stirrups, s, should be less than maximum spacing limit of shear reinforcement for end regions of beam smax,end ;

(3.46)

(3.47)

where, where, i = number of the beam (ith beam), Nbeam = total number of beams, k = three critical regions (left and right ends and the span) for shear design of beam, h = the height of beam section, Øb = the diameter of shear reinforcing bars (ties) .

Along the span of a beam member, the spacing between stirrups s should be less than maximum spacing of the shear reinforcement, smax,middle ;

(3.48)

(3.49)

where, where, i = number of the beam (ith beam), Nbeam = total number of beams, h = the height of beam section, bw and h = the width and height of column section, Asv = the area of shear reinforcement (tie), fy = the yielding strength of reinforcing steel.

The area of shear reinforcement (ties) in beam sections (span and ends of beam), Asv, should be greater than limitations on the minimum area of shear reinforcement, Asv,min ;

(3.50)

(3.51)

where, where, i = number of the beam (ith beam), Nbeam = total number of beams, k = three critical regions (left and right ends, and along the span) for shear design of a beam, fc = the compressive strength of concrete, fy = the yielding strength of reinforcing steel, bw and h = the width and height of beam section, s = the spacing between stirrups (ties) .

The width of beams, bw, should be greater than allowable minimum width for beams, bw,min ;

(3.52)

(3.53)

where, where, i = number of the beam (ith beam), Nbeam = total number of beams, Lbeam = the length of beam.

The height of beams, h, should be greater than allowable minimum height for beams, h,min ;

(3.54)

(3.55)

where, where, i = number of the beam (ith beam), Nbeam = total number of beams, Lbeam = length of the beam.

The detailed explanation of design philosophy of a beam member subject to ACI 318-05 specifications and the relevant constraints derived from these provisions are given in Appendix B.

## 3.4.3 Constraints for Joints

At frame joints, the width of beams, bw, should be smaller than the width of column, b, framed into the ends joint of beam ;

(3.56)

where, i = number of the joint (ith joint), Njoint = total number of joints.

The width of the top column, b, should be equal or smaller than the width of the bottom column at the column joints;

(3.57)

where, i = number of the joint (ith joint), Njoint = total number of joints.

The height of the top column, h, should be equal or smaller than the width of the bottom column at the column joints;

(3.58)

where, i = number of the joint (ith joint), Njoint = total number of joints.

The sum of nominal flexural strengths of columns framing into the joint, Î£Mnc, should be 1.2 times greater than the sum of nominal flexural strengths of beams framing into same joint, Î£Mnb,

(3.59)

where, i = number of the joint (ith joint), Njoint = total number of joints.

The relative story displacements, Î”, should be satisfy the requirements given in the design specification;

(3.60)

where, i = number of the story (ith story), Nstory = total number of stories, hi = the height of story from base level.

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