2. Limit state design of reinforced concrete T-section under bending

In accordance with EC-2 [13], the assumptions used at the limit state for the typical reinforced T-beam-cross section are, respectively, illustrated in Figure 1(a)–(c).

In the linear strain diagram of Figure 1b, the symbols ε<sup>s</sup> and εcu3 designate steel strain and the ultimate strain for the rectangular stress distribution compressive concrete design stress– strain relation. The parameter α represents the relative depth of the compressive concrete zone and the plastic neutral axis is located at the distance αd from the upper fiber for the ultimate limit state design, and x is the depth of elastic neutral axis for serviceability limit state design. In the assumed uniformly distributed stress diagram of Figure 1c, fcd is the design value of concrete compressive strength, γ<sup>c</sup> is the partial safety factor for concrete and fck is the characteristic compressive cylinder strength of ordinary or HSC at 28 days. In accordance with EC-2, the possibility of working with rectangular stress distribution is offered. This requires the

Design Optimization of Reinforced Ordinary and High-Strength Concrete Beams with Eurocode2 (EC-2) http://dx.doi.org/10.5772/intechopen.78734 123

Figure 1. (a) Typical T-beam cross section; (b) strains at ultimate limit state and (c) stresses at ultimate limit state.

introduction of a factor λ for the depth of the compression zone and a factor η for the design strength. The λ and η factors are both linearly dependent on the characteristic strength fck in accordance with the following Equations [13]:

$$
\lambda = 0.8 - \frac{\text{f}\_{\text{ck}} - 50}{400} \tag{1}
$$

$$
\mu = 1.0 - \frac{\text{f}\_{\text{ck}} - 50}{200} \tag{2}
$$

with 50 ≤ fck ≤ 90 MPa and λ = 0.8,η = 1.0 for fck ≤ 50 MPa.

Fc and Fs denote the resultants of internal forces in the HSC section and reinforcing steel, respectively.

The design yield strength of steel reinforcement is fyd = fyk/γ<sup>s</sup> where fyk is the characteristic elastic limit of steel and γ<sup>s</sup> is the partial safety factor. In addition, the steel strain is considered unlimited in accordance with the Eurocode2 provisions. In this chapter, for an optimal use of steel, the strain must always be greater or equal to elastic limit strain, εyd = fyd/Es where Es represents the elasticity modulus for steel.

#### 3. Formulation of the optimization problem

#### 3.1. Design variables

shaping and the construction details. However, the economy aspects in terms of costs and gain

Recent developments in the technology of materials have led to the use of the high-strength concrete (HSC); this is mainly due to its efficiency and economy. The reduction in the quantities of construction materials has enabled both a gain in weight reduction and in the foundation's cost. HSC has a high compressive strength in the range of 55–90 MPa; it not only has the advantage of reducing member size and story height, but also the volume of concrete and the area of formwork. In terms of the amount of steel reinforcement, there is a substantial difference between the normal-strength concrete structures compared to high-strength concrete structures [5, 6]. In this chapter, not only does it presents the minimum weight design but it presents a detailed objective function that considers the ratio cost not the absolute cost with sensitivity analysis of this cost ratio as well. It considers both shaping and material costs. The generalized reduced gradient (GRG) method is used to solve nonlinear programming problems. It is a very reliable and robust algorithm; also, various numerical methods have been

This work shows a method for minimizing separately the cost and weight of reinforced ordinary and high-strength concrete (HSC) T-beams at the limit state according to Eurocode2 (EC-2). The first objective function includes the costs of concrete, steel and formwork, whereas the second objective function represents the weight of the T-beam; all the constraints functions are set to meet the ultimate strength and serviceability requirements of Eurocode2 and current practices rules. The optimization process is developed through the use of the generalized reduced gradient algorithm. Two example problems are considered in order to illustrate the applicability of the proposed design model and solution methodology. It is concluded that this approach is economically more effective compared to conventional design methods applied by designers and engineers and can be extended to deal with other sections without major

2. Limit state design of reinforced concrete T-section under bending

T-beam-cross section are, respectively, illustrated in Figure 1(a)–(c).

In accordance with EC-2 [13], the assumptions used at the limit state for the typical reinforced

In the linear strain diagram of Figure 1b, the symbols ε<sup>s</sup> and εcu3 designate steel strain and the ultimate strain for the rectangular stress distribution compressive concrete design stress– strain relation. The parameter α represents the relative depth of the compressive concrete zone and the plastic neutral axis is located at the distance αd from the upper fiber for the ultimate limit state design, and x is the depth of elastic neutral axis for serviceability limit state design. In the assumed uniformly distributed stress diagram of Figure 1c, fcd is the design value of concrete compressive strength, γ<sup>c</sup> is the partial safety factor for concrete and fck is the characteristic compressive cylinder strength of ordinary or HSC at 28 days. In accordance with EC-2, the possibility of working with rectangular stress distribution is offered. This requires the

achieved should be the area where scope exists for extending the research works [1–4].

used in engineering optimization [7–12].

122 Optimum Composite Structures

alterations.

The design variables selected for the optimization are presented in Table 1.


(external moment ≤ resisting moment of the cross-section)

(internal force equilibrium)

(minimum steel percentage)

(maximum steel percentage)

Conditions on strain compatibility in steel:

plastic region at the ultimate limit state (ULS).)

(Compression reinforcement is not required.)

(external shear force ≤ resisting shear force)

b. Shear strength constraint:

c. Deflection constraint:

<sup>α</sup> <sup>¼</sup> fyd

fcd AS

ηλbwd 

As

As bwh þ ð Þ b � bw hf

which ensures that the section behaves as the T-beam section shown in Figure 1a.

1 α 

εcu3

VEd ≤VRd,max ¼ ν<sup>1</sup>

Ic <sup>¼</sup> bwh<sup>3</sup>

5wL<sup>4</sup> 384 EcmIc

<sup>3</sup> <sup>þ</sup> ð Þ <sup>b</sup> � bw <sup>h</sup><sup>3</sup>

In Eqs. (7) and (8) above, it is assumed that the neutral axis position is under the beam flange

� 1 <sup>≥</sup>

(In the case of Pivot B, optimal use of steel requires that strains in steel must be limited to

f yd Es

f cdbwz

λαð Þ 1–0; 5λα ≤ μlimit (11)

tg ð Þþ <sup>θ</sup> cotgð Þ <sup>θ</sup> (12)

≤ δlim (13)

<sup>3</sup> <sup>þ</sup> nAsd<sup>2</sup> � Ahx<sup>2</sup> (14)

Ah ¼ bwh þ ð Þ b � bw hf þ nAs (15)

� ð Þ <sup>b</sup> � bw hf

Design Optimization of Reinforced Ordinary and High-Strength Concrete Beams with Eurocode2 (EC-2)

<sup>λ</sup>bwd (7)

http://dx.doi.org/10.5772/intechopen.78734

≤ pmax (9)

(10)

125

bwd <sup>≥</sup> pmin (8)

Table 1. Definition of design variables.

#### 3.2. Objective functions

#### 3.2.1. Cost function

The objective function to be minimized in the optimization problems is the total cost of construction material per unit length of the beam. This function can be defined as:

$$\mathbf{C}\_{0}/\mathbf{L} = \mathbf{C}\_{c}(\mathbf{b}\_{\mathbf{w}}\mathbf{h} + (\mathbf{b} - \mathbf{b}\_{\mathbf{w}})\mathbf{h}\_{\mathbf{f}}) + \mathbf{C}\_{s}\mathbf{A}\_{s} + \mathbf{C}\_{\mathbf{f}}[\mathbf{b} + 2\mathbf{h}] \to \text{Minimum} \tag{3}$$

Thus, the cost function to be minimized can be written as follows:

$$\mathbf{C} = \frac{\mathbf{C\_O}}{\mathbf{C\_t L}} = \mathbf{b\_w}\mathbf{h} + (\mathbf{b} - \mathbf{b\_w})\mathbf{h\_f} + \left(\frac{\mathbf{C\_s}}{\mathbf{C\_c}}\right)\mathbf{A\_s} + \left(\frac{\mathbf{C\_f}}{\mathbf{C\_c}}\right)[\mathbf{b} + 2\mathbf{h}] \to \text{Minimum} \tag{4}$$

The values of the cost ratios Cs/Cc and Cf/Cc vary from one country to another and may eventually vary from one region to another for certain countries [14, 15].

#### 3.2.2. Weight function

The weight function to be minimized can be written as follows:

$$\mathbf{W} = (\mathbf{b}\_{\mathbf{w}}\mathbf{h} + (\mathbf{b} - \mathbf{b}\_{\mathbf{w}})\mathbf{h}\_{\mathbf{f}})\rho \longrightarrow (\text{minimum}) \tag{5}$$

where

r is the density of the reinforced concrete T-beams and W is the unit weight per unit length of the reinforced concrete T- beams.

#### 3.3. Design constraints

#### a. Behavior constraints:

$$\mathbf{M}\_{\rm Ed} \le \eta \mathbf{f}\_{\rm cd} (\mathbf{b} - \mathbf{b}\_{\rm w}) \mathbf{h}\_{\rm l} (\mathbf{d} - \mathbf{0}, 50 \mathbf{h}\_{\rm l}) + \eta \lambda \mathbf{f}\_{\rm cd} \mathbf{b}\_{\rm w} \mathbf{d}^2 \mathbf{a} (\mathbf{1} - \mathbf{0}, 5 \lambda \mathbf{a}) \tag{6}$$

Design Optimization of Reinforced Ordinary and High-Strength Concrete Beams with Eurocode2 (EC-2) http://dx.doi.org/10.5772/intechopen.78734 125

(external moment ≤ resisting moment of the cross-section)

$$\mathbf{a} = \left(\frac{\mathbf{f}\_{\rm yd}}{\mathbf{f}\_{\rm cd}}\right) \left(\frac{\mathbf{A}\_{\rm S}}{\eta \lambda \mathbf{b}\_{\rm w} \mathbf{d}}\right) - \frac{(\mathbf{b} - \mathbf{b}\_{\rm w}) \mathbf{h}\_{\rm f}}{\lambda \mathbf{b}\_{\rm w} \mathbf{d}} \tag{7}$$

(internal force equilibrium)

$$\frac{\mathbf{A}\_s}{\mathbf{b}\_\mathbf{w}\mathbf{d}} \ge \mathbf{p}\_{\min} \tag{8}$$

(minimum steel percentage)

$$\frac{\text{As}}{\text{lb}\_{\text{w}}\text{h} + (\text{b} - \text{b}\_{\text{w}})\text{h}\_{\text{f}}} \leq \text{p}\_{\text{max}}\tag{9}$$

(maximum steel percentage)

3.2. Objective functions

Table 1. Definition of design variables.

<sup>C</sup> <sup>¼</sup> CO

the reinforced concrete T- beams.

3.3. Design constraints

a. Behavior constraints:

3.2.2. Weight function

where

The objective function to be minimized in the optimization problems is the total cost of

Cs Cc 

The values of the cost ratios Cs/Cc and Cf/Cc vary from one country to another and may

r is the density of the reinforced concrete T-beams and W is the unit weight per unit length of

MEd <sup>≤</sup> <sup>η</sup>fcdð Þ <sup>b</sup> � bw hfð Þþ <sup>d</sup> � <sup>0</sup>; 50hf ηλfcdbwd2

C0=L ¼ Ccðbwh þ ð Þ b � bw hfÞ þ CsAs þ Cf½ �! b þ 2h Minimum (3)

Cf Cc 

<sup>W</sup> <sup>¼</sup> ð Þ bwh <sup>þ</sup> ð Þ <sup>b</sup> � bw hf <sup>r</sup> !ð Þ minimum (5)

½ �! b þ 2h Minimum (4)

αð Þ 1–0; 5λα (6)

As þ

construction material per unit length of the beam. This function can be defined as:

α Relative depth of compressive concrete zone

Thus, the cost function to be minimized can be written as follows:

eventually vary from one region to another for certain countries [14, 15].

CcL <sup>¼</sup> bwh <sup>þ</sup> ð Þ <sup>b</sup> � bw hf <sup>þ</sup>

Design variables Defined variables

bw Web width h Total depth d Effective depth hf Flange depth

b Effective width of compressive flange

As Area of tension reinforcement

The weight function to be minimized can be written as follows:

3.2.1. Cost function

124 Optimum Composite Structures

In Eqs. (7) and (8) above, it is assumed that the neutral axis position is under the beam flange which ensures that the section behaves as the T-beam section shown in Figure 1a.

Conditions on strain compatibility in steel:

$$
\varepsilon\_{\rm cu3} \left( \left( \frac{1}{\alpha} \right) - 1 \right) \gtrsim \frac{f\_{yd}}{E\_{\rm s}} \tag{10}
$$

(In the case of Pivot B, optimal use of steel requires that strains in steel must be limited to plastic region at the ultimate limit state (ULS).)

$$
\lambda \alpha (1 - 0, 5 \lambda \alpha) \le \mu\_{\text{limit}} \tag{11}
$$

(Compression reinforcement is not required.)

b. Shear strength constraint:

$$V\_{Ed} \le V\_{Rd, \max} = \nu\_1 \frac{f\_{cl} b\_w z}{t g\left(\theta\right) + c \text{tg}\left(\theta\right)}\tag{12}$$

(external shear force ≤ resisting shear force)

c. Deflection constraint:

$$\frac{5\upsilon L^4}{384\ E\_{cm}I\_c} \le \delta\_{lim}\tag{13}$$

$$I\_c = \frac{b\_w h^3}{3} + \frac{(b - b\_w)h^3}{3} + nA\_s d^2 - A\_h x^2 \tag{14}$$

$$A\_{\rm h} = b\_w h + (b - b\_w) h\_f + n A\_s \tag{15}$$

$$\chi = \frac{\frac{b\_v h^2}{2} + \frac{(b - b\_w)h\_f^2}{2} + nA\_s d}{A\_h} \tag{16}$$

#### d. Geometric design variable constraints including rules of current practice:

$$h \ge \frac{L}{16} \tag{17}$$

3.6. Solution methodology: Generalized reduced gradient method

or constraints is nonlinear.

Solver is GRG.

can be found in [19].

EC-2 design code.

4. Numerical results and discussion

4.1. Design example A for reinforced HSC T-beams

The corresponding preassigned parameters are defined as follows:

The objective function Eq. (4), the objective function Eq. (5) and the constraints equations, Eq. (6) through Eq.(23), together form a nonlinear optimization problem. The reasons for the nonlinearity of this optimization problem are essentially due to the expressions of the crosssectional area, bending moment capacity and other constraints equations. Both the objective function and the constraint functions are nonlinear in terms of the design variables. In order to solve this nonlinear optimization problem, the generalized reduced gradient (GRG) algorithm is used. This algorithm was first developed in late 1960 by Jean Abadie [16] as an extension of the reduced gradient method and then since has been refined by several other researchers [17, 18]. GRG nonlinear should be selected if any of the equations involving decision variables

Design Optimization of Reinforced Ordinary and High-Strength Concrete Beams with Eurocode2 (EC-2)

http://dx.doi.org/10.5772/intechopen.78734

127

Microsoft Excel, beginning with version 3.0 in 1991, incorporates an NLP solver that operates on values and formulas of a spreadsheet model. Version 4.0 and later include LP solver and mixed-integer programming (MIP) capability for both linear and nonlinear problems. The Microsoft Office Excel Solver tool uses several algorithms to find optimal solutions. The GRG nonlinear solving method for nonlinear optimization uses the Generalized Reduced Gradient code. The Simplex LP solving method for linear programming uses the Simplex and dual Simplex method. The Evolutionary solving method for non-smooth optimization uses a variety of genetic algorithm and local search methods. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formulae cells whose values are to be constrained (the constraints) and a formula cell designated as the optimization objective. The solver uses the spreadsheet interpreter to evaluate the constraint and objective functions and approximates derivatives, using finite differences. The NLP solution engine for the Excel

The generalized reduced gradient method is applied as it has the following advantages: (i) the GRG method is widely recognized as an efficient method for solving a relatively wide class of nonlinear optimization problems; (ii) the program can handle up to 200 constraints, which is suitable for reinforced ordinary and HSC beam design optimization problems; and (iii) GRG transforms inequality constraints into equality constraints by introducing slack variables. Hence all the constraints are of equality form. A more detailed description of the GRG method

The numerical example A corresponds to a high-strength concrete T-beam belonging to a bridge deck, simply supported at its ends and predesigned in accordance with provisions of

$$\frac{d}{h} = 0.90\tag{18}$$

$$0.20 \le \frac{b\_w}{d} \le 0.50\tag{19}$$

$$\frac{\left(b - b\_{w}\right)}{2} \le \frac{L}{10} \tag{20}$$

$$\frac{b}{h\_f} \le 8 \tag{21}$$

$$h\_f \gtrsim h\_{f \min} \tag{22}$$

$$\frac{b}{b\_w} \ge 3\tag{23}$$

where:

μlimit is the limit value of the reduced moment.

θ is the angle between concrete compression struts and the main chord

ν<sup>1</sup> is a nondimensional coefficient, ν<sup>1</sup> = 0.60(1-fck/250);

z is the lever arm, z = 0.9d;

hfmin is the minimum depth of flange.

#### 3.4. Optimization based on minimum cost design

The optimum cost design of reinforced concrete T-beams under the limit state can be stated as follows:

For given material properties, loading data and constant parameters, find the design variables defined in Table 1 that minimize the cost function defined in Eq. (4) subjected to the design constraints given in Eq. (6) through Eq. (23).

#### 3.5. Optimization based on minimum weight design

Find the design variables that minimize total weight per unit length defined in Eq. (5), subjected to the design constraints given in Eq. (6) through Eq. (23).

## 3.6. Solution methodology: Generalized reduced gradient method

x ¼

d. Geometric design variable constraints including rules of current practice:

bwh<sup>2</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>b</sup>�bw <sup>h</sup><sup>2</sup>

h ≥ L

bw

L

d

0:20 ≤

ð Þ b � bw <sup>2</sup> <sup>≤</sup>

> b hf

b bw

The optimum cost design of reinforced concrete T-beams under the limit state can be stated as

For given material properties, loading data and constant parameters, find the design variables defined in Table 1 that minimize the cost function defined in Eq. (4) subjected to the design

Find the design variables that minimize total weight per unit length defined in Eq. (5),

where:

126 Optimum Composite Structures

follows:

μlimit is the limit value of the reduced moment.

z is the lever arm, z = 0.9d;

hfmin is the minimum depth of flange.

ν<sup>1</sup> is a nondimensional coefficient, ν<sup>1</sup> = 0.60(1-fck/250);

3.4. Optimization based on minimum cost design

constraints given in Eq. (6) through Eq. (23).

3.5. Optimization based on minimum weight design

subjected to the design constraints given in Eq. (6) through Eq. (23).

θ is the angle between concrete compression struts and the main chord

f <sup>2</sup> þ nAsd

<sup>16</sup> (17)

<sup>d</sup> <sup>≤</sup> <sup>0</sup>:<sup>50</sup> (19)

≤ 8 (21)

≥ 3 (23)

hf ≥ hfmin (22)

<sup>10</sup> (20)

<sup>h</sup> <sup>¼</sup> <sup>0</sup>:<sup>90</sup> (18)

(16)

Ah

The objective function Eq. (4), the objective function Eq. (5) and the constraints equations, Eq. (6) through Eq.(23), together form a nonlinear optimization problem. The reasons for the nonlinearity of this optimization problem are essentially due to the expressions of the crosssectional area, bending moment capacity and other constraints equations. Both the objective function and the constraint functions are nonlinear in terms of the design variables. In order to solve this nonlinear optimization problem, the generalized reduced gradient (GRG) algorithm is used. This algorithm was first developed in late 1960 by Jean Abadie [16] as an extension of the reduced gradient method and then since has been refined by several other researchers [17, 18]. GRG nonlinear should be selected if any of the equations involving decision variables or constraints is nonlinear.

Microsoft Excel, beginning with version 3.0 in 1991, incorporates an NLP solver that operates on values and formulas of a spreadsheet model. Version 4.0 and later include LP solver and mixed-integer programming (MIP) capability for both linear and nonlinear problems. The Microsoft Office Excel Solver tool uses several algorithms to find optimal solutions. The GRG nonlinear solving method for nonlinear optimization uses the Generalized Reduced Gradient code. The Simplex LP solving method for linear programming uses the Simplex and dual Simplex method. The Evolutionary solving method for non-smooth optimization uses a variety of genetic algorithm and local search methods. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formulae cells whose values are to be constrained (the constraints) and a formula cell designated as the optimization objective. The solver uses the spreadsheet interpreter to evaluate the constraint and objective functions and approximates derivatives, using finite differences. The NLP solution engine for the Excel Solver is GRG.

The generalized reduced gradient method is applied as it has the following advantages: (i) the GRG method is widely recognized as an efficient method for solving a relatively wide class of nonlinear optimization problems; (ii) the program can handle up to 200 constraints, which is suitable for reinforced ordinary and HSC beam design optimization problems; and (iii) GRG transforms inequality constraints into equality constraints by introducing slack variables. Hence all the constraints are of equality form. A more detailed description of the GRG method can be found in [19].
