4. Numerical results and discussion

#### 4.1. Design example A for reinforced HSC T-beams

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 EC-2 design code.

The corresponding preassigned parameters are defined as follows:

L = 25 m; MEd = 1.35 MG + 1.5 MQ = 9 MNm; VEd = 1.35 VG + 1.5 VQ = 3.1 MN.

w = 0.60MN/ml (the total distribution load (dead load + live load)), δlim = L/250 = 0.100 m.

The optimal solutions using the minimum cost design and the minimum weight design are

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It is shown from Table 2 that the gain and optimum values for minimum cost design and for

From the above results, it is clearly shown that significant cost saving of the order of 47% can be obtained using the proposed minimum weight design formulation and 22% through the use

In this section, the optimal solution is obtained according to practical consideration: (i) the total depth is imposed, h = himposed; (ii) the effective width of compressive flange is imposed, b=bimposed; (iii) the reinforcing steel is imposed, As = Asimposed; and (4i) the flange depth is

The gain depends on the type of formwork used. We distinguish the wood formwork:

Further practical requirements can also be implemented, such as esthetic, architectural and limited authorized templates. The optimal solutions obtained using the particular conditions

From the above results, it is clearly seen that a significant cost saving between 08% and 23%

The relative gains can be determined for various values of unit-cost ratios: Cs/Cc = 10; 20; 30;

; S500 and C70/85 22

Table 3. The variation of relative gain with particular conditions imposed such as the HSC T-beam dimensions and

shown in Table 2.

4.1.2. Parametric study

imposed, hf = hfimposed.

imposed are shown in Table 3.

4.1.3. Sensitivity analysis

Imposed reinforcement As ≤ 0.0150 m<sup>2</sup>

reinforcing steel.

minimum weight design are different.

of minimum cost-design approach.

Cf/Cc = 0.01 and the steel formwork Cf/Cc = 0.10.

can be obtained by using this parametric study.

40; 50; 60; 70; 80; 90; 100 for a given unit cost ratio Cf/Cc = 0.01

The corresponding results are reported in Table 4 and represented in Figure 2.

Optimal solution with. Gain (%)

Classes(S500, C70/85); Cs/Cc = 40; Cf/Cc = 0.01wood formwork 22 Classes(S500, C70/85); Cs/Cc = 40; Cf/Cc = 0.10 steel formwork 19 Classes(S500,C70/85) and Cf/Cc = 0 the cost of the formwork is negligible 23 Classes(S400,C70/85); Cs/Cc = 40; Cf/Cc = 0.01wood formwork 08 Imposed height h = 1.70 m; S500 and C70/85 21 Imposed width b = 1.00 m; S500 and C70/85 22

Imposed flange depth hf = 0.10 m; S500 andC70/85 22

Input data for HSC characteristics:

C70/85; fck = 70 MPa; γ<sup>c</sup> = 1.5; fcd = 46.67 MPa; r = 0.025 MN/m<sup>3</sup> ; Ecm = 40,743 MPa;

λ = 0.75; η = 0.90; εcu3(‰) = 2.7; εc3(‰) = 2.4; hfmin = 0.10 m; fctm = 4.6 MPa;

μlimit = 0.329; αlimit = 0.554 for S500 and C70/85.

Input data for steel characteristics:

S500; fyk = 500 MPa; γ<sup>s</sup> = 1.15; fyd = fyk/γ<sup>s</sup> = 435 MPa; n = 15;

S400; fyk = 400 MPa; γ<sup>s</sup> = 1.15; fyd = fyk/γ<sup>s</sup> = 348 MPa; fyd/fcd = 9.32 for classes (S500, C70/85);

fyd/fcd = 7.46 for classes (S400, C70/85); μlimit = 0.352; αlimit = 0.6081 for S400 and C70/85;

$$\mathbf{E\_s = 2 \times 10^5 \text{ MPa}; \mathbf{p\_{min} = 0.26 \text{ f\_{ctm}/f\_{yk} = 0.002392; } \mathbf{p\_{max} = 4\%...}}$$

Input data for units cost ratios of construction materials:

Cs/Cc = 40 for HSC concrete;

Cf/Cc = 0.01 for wood formwork;

Cf/Cc = 0.10 for metal formwork;

Cf/Cc = 0.00 in the case of the cost of the formwork is negligible.

#### 4.1.1. Comparison between the minimum cost design and the minimum weight design of HSC T-beams

The vector of design variables including the geometric dimensions of the T-beam cross-section and the area of tension reinforcement as obtained from the standard design approach solution and the optimal cost design solution using the proposed approach is shown in Table 2.


Table 2. Comparison of the optimal solutions with minimum weight and minimum cost design for HSC.

The optimal solutions using the minimum cost design and the minimum weight design are shown in Table 2.

It is shown from Table 2 that the gain and optimum values for minimum cost design and for minimum weight design are different.

From the above results, it is clearly shown that significant cost saving of the order of 47% can be obtained using the proposed minimum weight design formulation and 22% through the use of minimum cost-design approach.

## 4.1.2. Parametric study

L = 25 m; MEd = 1.35 MG + 1.5 MQ = 9 MNm; VEd = 1.35 VG + 1.5 VQ = 3.1 MN.

λ = 0.75; η = 0.90; εcu3(‰) = 2.7; εc3(‰) = 2.4; hfmin = 0.10 m; fctm = 4.6 MPa;

C70/85; fck = 70 MPa; γ<sup>c</sup> = 1.5; fcd = 46.67 MPa; r = 0.025 MN/m<sup>3</sup>

S500; fyk = 500 MPa; γ<sup>s</sup> = 1.15; fyd = fyk/γ<sup>s</sup> = 435 MPa; n = 15;

Es = 2 105 MPa; pmin = 0.26 fctm/fyk = 0.002392; pmax = 4%. Input data for units cost ratios of construction materials:

Cf/Cc = 0.00 in the case of the cost of the formwork is negligible.

formwork

b(m) 1.20 0.86 0.52 bw(m) 0.40 0.28 0.28 h(m) 1.40 1.58 1.56 d(m) 1.26 1.42 1.40 hf(m) 0.15 0.11 0.10

α 0.554 0.342 0.554 Gain 22% 47%

) 185x10<sup>4</sup> 161x10<sup>4</sup> 181 x10<sup>4</sup>

Table 2. Comparison of the optimal solutions with minimum weight and minimum cost design for HSC.

Input data for HSC characteristics:

128 Optimum Composite Structures

Input data for steel characteristics:

Cs/Cc = 40 for HSC concrete;

Design variables vector.

AS(m2

Cf/Cc = 0.01 for wood formwork; Cf/Cc = 0.10 for metal formwork;

> Initial design

μlimit = 0.329; αlimit = 0.554 for S500 and C70/85.

w = 0.60MN/ml (the total distribution load (dead load + live load)), δlim = L/250 = 0.100 m.

S400; fyk = 400 MPa; γ<sup>s</sup> = 1.15; fyd = fyk/γ<sup>s</sup> = 348 MPa; fyd/fcd = 9.32 for classes (S500, C70/85);

4.1.1. Comparison between the minimum cost design and the minimum weight design of HSC T-beams The vector of design variables including the geometric dimensions of the T-beam cross-section and the area of tension reinforcement as obtained from the standard design approach solution and the optimal cost design solution using the proposed approach is shown in Table 2.

> Optimal solution with minimum cost (S500, C70/85), Cs/Cc = 40, Cf/Cc = 0.01 wood

fyd/fcd = 7.46 for classes (S400, C70/85); μlimit = 0.352; αlimit = 0.6081 for S400 and C70/85;

; Ecm = 40,743 MPa;

Optimal solution with minimum

weight

In this section, the optimal solution is obtained according to practical consideration: (i) the total depth is imposed, h = himposed; (ii) the effective width of compressive flange is imposed, b=bimposed; (iii) the reinforcing steel is imposed, As = Asimposed; and (4i) the flange depth is imposed, hf = hfimposed.

The gain depends on the type of formwork used. We distinguish the wood formwork: Cf/Cc = 0.01 and the steel formwork Cf/Cc = 0.10.

Further practical requirements can also be implemented, such as esthetic, architectural and limited authorized templates. The optimal solutions obtained using the particular conditions imposed are shown in Table 3.

From the above results, it is clearly seen that a significant cost saving between 08% and 23% can be obtained by using this parametric study.

## 4.1.3. Sensitivity analysis

The relative gains can be determined for various values of unit-cost ratios: Cs/Cc = 10; 20; 30; 40; 50; 60; 70; 80; 90; 100 for a given unit cost ratio Cf/Cc = 0.01


The corresponding results are reported in Table 4 and represented in Figure 2.

Table 3. The variation of relative gain with particular conditions imposed such as the HSC T-beam dimensions and reinforcing steel.


Table 4. Variation of relative gain in percentage (%) versus unit cost ratio Cs/Cc for a given cost ratio Cf/Cc = 0.01.

It is shown in Table 4 and Figure 2 that the relative gain decreases for increasing values of the unit cost ratio Cs/Cc, stabilizes around an average value for 40 ≤ Cs/Cc ≤ 60 and then increases significantly beyond this average value for a given cost ratio Cf/Cc = 0.01.

From Table 5 and Figure 3, the gain decreases monotonically with the increase of unit cost

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Table 5. Variation of relative gain in percentage (%) versus unit cost ratio Cf/Cc for a given cost ratio Cs/Cc = 40.

The numerical example B corresponds to a concrete T-beam belonging to a pedestrian deck, simply supported at its ends and predesigned in accordance with the provisions of EC-2

Figure 3. Variation of relative gain in percentage (%) versus unit cost ratio Cf/Cc for a given cost ratio Cs/Cc = 40.

ratio Cf/Cc for a given cost ratio Cs/Cc = 40.

design code.

4.2. Design example B for reinforced ordinary concrete T-beams

(S500; C70/85) Cs/Cc = 40 Cf/Cc Gain (%) 0.01 22 0.02 21 0.03 21 0.04 20 0.05 20 0.06 19 0.07 19 0.08 19 0.09 19 0.10 18

The relative gains can be determined for various values of unit cost ratios: Cf/Cc = 0.01; 0.02; 0.03; 0.04; 0.05; 0.06; 0.07; 0.08; 0.08; 0.09; 0.10 for a given unit cost ratio Cs/Cc = 40.

The corresponding results are reported in Table 5 and presented in Figure 3.

Figure 2. Variation of relative gain in percentage (%) versus unit cost ratio Cs/Cc for a given cost ratio Cf/Cc = 0.01.


Table 5. Variation of relative gain in percentage (%) versus unit cost ratio Cf/Cc for a given cost ratio Cs/Cc = 40.

From Table 5 and Figure 3, the gain decreases monotonically with the increase of unit cost ratio Cf/Cc for a given cost ratio Cs/Cc = 40.

#### 4.2. Design example B for reinforced ordinary concrete T-beams

It is shown in Table 4 and Figure 2 that the relative gain decreases for increasing values of the unit cost ratio Cs/Cc, stabilizes around an average value for 40 ≤ Cs/Cc ≤ 60 and then increases

Table 4. Variation of relative gain in percentage (%) versus unit cost ratio Cs/Cc for a given cost ratio Cf/Cc = 0.01.

The relative gains can be determined for various values of unit cost ratios: Cf/Cc = 0.01; 0.02;

Figure 2. Variation of relative gain in percentage (%) versus unit cost ratio Cs/Cc for a given cost ratio Cf/Cc = 0.01.

significantly beyond this average value for a given cost ratio Cf/Cc = 0.01.

(S500; C70/85) Cf/Cc = 0.01 Gain (%) 10 33 20 27 30 24 40 22 50 22 60 22 70 23 80 24 90 26 100 27

130 Optimum Composite Structures

0.03; 0.04; 0.05; 0.06; 0.07; 0.08; 0.08; 0.09; 0.10 for a given unit cost ratio Cs/Cc = 40.

The corresponding results are reported in Table 5 and presented in Figure 3.

The numerical example B corresponds to a concrete T-beam belonging to a pedestrian deck, simply supported at its ends and predesigned in accordance with the provisions of EC-2 design code.

Figure 3. Variation of relative gain in percentage (%) versus unit cost ratio Cf/Cc for a given cost ratio Cs/Cc = 40.

The preassigned parameters are defined as follows:

$$\text{mL} = 20 \text{ m}; \text{M}\_{\text{Ed}} = 5 \text{M} \text{N} \text{m}; \text{V}\_{\text{Ed}} = 1.1 \text{M} \text{N}; \text{w} = 0.043 \text{M} \text{N}/\text{ml}; \text{S}\_{\text{lim}} = \text{L}/250 = 0.080 \text{ m}.$$

Design variables vector

AS(m2

Initial design, C20/25 & S400

Imposed reinforcementAs ≤ 0.0120 m<sup>2</sup>

and weight.

Optimal solution with minimum

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cost, C20/25 & S400

weight, C20/25 & S400

) <sup>125</sup> <sup>10</sup><sup>4</sup> <sup>123</sup> <sup>10</sup><sup>4</sup> <sup>122</sup> <sup>10</sup><sup>4</sup>

Table 6. Comparison of the optimal solutions with minimum weight and minimum cost design.

Optimal solution with Gain (%) fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01 wood formwork, C20/25 & S400 14 fyd/fcd = 38.39; Cs/Cc = 30; Cf/Cc = 0.01wood formwork, C20/25 & S500 09 fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.00; C20/25 & S400 15 Imposed web with bw = 0.30 m; fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01; C20/25 & S400 13

Imposed height h = 1.50 m; fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01; C20/25 & S400 11 Imposed relative depthα = 0.600; fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01; C20/25 & S400 14

(S400; C20/25) Cf/Cc = 0.01 Gain (%) 10 18 20 16 30 14 40 13 50 12 60 12 70 12 80 11 90 11 100 11

Table 7. Variation of relative gain with particular conditions imposed such as the T-beam dimensions, reinforcing steel

Table 8. Variation of relative gain in percentage (%) versus unit cost ratio Cs/Cc for a given cost ratio Cf/Cc = 0.01.

; fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01; C20/25 & S400 14

b(m) 1.20 1.30 1.25 bw(m) 0.40 0.28 0.29 h(m) 1.60 1.57 1.60 d(m) 1.44 1.41 1.44 hf(m) 0.14 0.17 0.16

α 0.668 0.668 0.668 C 1.171 1.0281 Gain 23% 14%

Input data for ordinary concrete characteristics:

$$\text{1C20/25; f}\_{\text{ck}} = 20 \text{ MPa; } \gamma\_{\text{c}} = 1.5; \text{f}\_{\text{cd}} = 11.33 \text{ MPa; } \rho = 0.025 \text{MN/m}^3; \text{E}\_{\text{cm}} = 30,000 \text{ MPa;}$$

λ = 0.80; η =1.00; εcu3(‰) = 2; εc3(‰) = 3.5; hfmin = 0.15 m; fctm = 2.20 MPa; n = 15;

μlimit = 0.372; αlimit = 0.6167 for S500 and C20/25.

μlimit = 0.392; αlimit = 0.6680 for S400; and C20/25.

Input data for steel characteristics:

S400; fyk = 400 MPa; γ<sup>s</sup> = 1.15; fyd = fyk/γ<sup>s</sup> = 348 MPa;

Es = 2 105 MPa; pmin = 0.26 fctm/fyk = 0.00143; pmax = 4%;

fyd/fcd = 30.71 for classes (S400, C20/25);

fyd/fcd = 38.39 for classes (S500, C20/25).

Input data for units cost ratios of construction materials:

Cs/Cc = 30 for ordinary concrete.

Cf/Cc = 0.10 for metal formwork.

Cf/Cc = 0.01 for wood formwork.

4.2.1. Comparison between the minimum cost design and the minimum weight design of ordinary concrete T-beams

The optimal solutions using the minimum weight design and the minimum cost design are shown in Table 6.

It is shown in Table 6 that the gain and the optimum values for minimum weight design and for minimum cost design are different.

From the above results, it is clearly shown that a significant cost saving of the order of 23% can be obtained using the proposed minimum weight design formulation and 14% through the use of the minimum cost design approach.

#### 4.2.2. Parametric study

In this section, the optimal solution is obtained through the considerations: (i) one of the dimensions of HSC T-section is imposed, h = 1.50 m; (ii) the imposed reinforcing steel As = <sup>120</sup> <sup>10</sup><sup>4</sup> m2 ; (iii) imposed web width bW = 0.30 m; and (iv) imposed relative depth of compressive concrete zone α = 0.6000

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Table 6. Comparison of the optimal solutions with minimum weight and minimum cost design.

The preassigned parameters are defined as follows:

Input data for ordinary concrete characteristics:

μlimit = 0.372; αlimit = 0.6167 for S500 and C20/25.

μlimit = 0.392; αlimit = 0.6680 for S400; and C20/25.

S400; fyk = 400 MPa; γ<sup>s</sup> = 1.15; fyd = fyk/γ<sup>s</sup> = 348 MPa;

Es = 2 105 MPa; pmin = 0.26 fctm/fyk = 0.00143; pmax = 4%;

Input data for units cost ratios of construction materials:

Input data for steel characteristics:

132 Optimum Composite Structures

fyd/fcd = 30.71 for classes (S400, C20/25); fyd/fcd = 38.39 for classes (S500, C20/25).

Cs/Cc = 30 for ordinary concrete. Cf/Cc = 0.10 for metal formwork. Cf/Cc = 0.01 for wood formwork.

for minimum cost design are different.

of the minimum cost design approach.

compressive concrete zone α = 0.6000

concrete T-beams

shown in Table 6.

4.2.2. Parametric study

<sup>120</sup> <sup>10</sup><sup>4</sup> m2

C20/25; fck = 20 MPa; γ<sup>c</sup> = 1.5;fcd = 11.33 MPa; r = 0.025MN/m<sup>3</sup>

L = 20 m; MEd = 5MNm; VEd = 1.1MN; w = 0.043MN/ml; δlim = L/250 = 0.080 m.

λ = 0.80; η =1.00; εcu3(‰) = 2; εc3(‰) = 3.5; hfmin = 0.15 m; fctm = 2.20 MPa; n = 15;

4.2.1. Comparison between the minimum cost design and the minimum weight design of ordinary

The optimal solutions using the minimum weight design and the minimum cost design are

It is shown in Table 6 that the gain and the optimum values for minimum weight design and

From the above results, it is clearly shown that a significant cost saving of the order of 23% can be obtained using the proposed minimum weight design formulation and 14% through the use

In this section, the optimal solution is obtained through the considerations: (i) one of the dimensions of HSC T-section is imposed, h = 1.50 m; (ii) the imposed reinforcing steel As =

; (iii) imposed web width bW = 0.30 m; and (iv) imposed relative depth of

; Ecm = 30,000 MPa;


Table 7. Variation of relative gain with particular conditions imposed such as the T-beam dimensions, reinforcing steel and weight.


Table 8. Variation of relative gain in percentage (%) versus unit cost ratio Cs/Cc for a given cost ratio Cf/Cc = 0.01.

Figure 4. Variation of relative gain in percentage (%) versus unit cost ratio Cs/Cc for a given cost ratio Cf/Cc = 0.01.


The corresponding results are reported in Table 8 and presented graphically in Figure 4.

Figure 5. Variation of relative gain in percentage (%) versus unit cost ratio Cf/Cc for a given cost ratio Cs/Cc = 30.

0.03; 0.04; 0.05; 0.06; 0.07; 0.08; 0.08; 0.09; 0.10 for a given unit cost ratio Cs/Cc = 30.

The following important conclusions are drawn on the basis of this chapter:

reduced number of independent design variables.

the objective function and not by the absolute cost values.

unit cost ratio Cs/Cc for a given value of Cf/Cc = 0.01.

ratio Cf/Cc for a given value of Cs/Cc = 30.

cross-section dimensions.

5. Conclusions

It is shown in Table 8 and Figure 4 that the relative gain decreases for increasing values of the

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The relative gains can be determined for various values of the unit cost ratios: Cf/Cc = 0.01; 0.02;

From Table 9 and Figure 5, the gain decreases monotonically with the increase of unit cost

• The problem formulation of the optimal cost design of reinforced concrete T-beams can be cast into a nonlinear programming problem; the numerical solution is efficiently determined using the GRG method in a space of only a few variables representing the concrete

• The space of feasible design solutions and the optimal solutions can be obtained from a

• The optimal values of the design variables are only affected by the relative cost values of

The corresponding results are reported in Table 9 and illustrated graphically in Figure 5.

Table 9. Variation of relative gain in percentage (%) versus unit cost ratio Cf/Cc for Cs/Cc = 30.

Further practical requirements can also be implemented, such as esthetic, architectural and limited authorized template.

The optimal solutions obtained using the particular conditions imposed are shown in Table 7.

From the above results, it is clearly seen that a significant cost saving between 09 and 15% can be obtained by using this parametric study.

#### 4.2.3. Sensitivity analysis

The relative gains can be determined for various values of the unit cost ratios: Cs/Cc = 10; 20; 30; 40; 50; 60; 70; 80; 90; 100 for a given unit cost ratio Cf/Cc = 0.01

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Figure 5. Variation of relative gain in percentage (%) versus unit cost ratio Cf/Cc for a given cost ratio Cs/Cc = 30.

The corresponding results are reported in Table 8 and presented graphically in Figure 4.

It is shown in Table 8 and Figure 4 that the relative gain decreases for increasing values of the unit cost ratio Cs/Cc for a given value of Cf/Cc = 0.01.

The relative gains can be determined for various values of the unit cost ratios: Cf/Cc = 0.01; 0.02; 0.03; 0.04; 0.05; 0.06; 0.07; 0.08; 0.08; 0.09; 0.10 for a given unit cost ratio Cs/Cc = 30.

The corresponding results are reported in Table 9 and illustrated graphically in Figure 5.

From Table 9 and Figure 5, the gain decreases monotonically with the increase of unit cost ratio Cf/Cc for a given value of Cs/Cc = 30.

#### 5. Conclusions

Further practical requirements can also be implemented, such as esthetic, architectural and

Figure 4. Variation of relative gain in percentage (%) versus unit cost ratio Cs/Cc for a given cost ratio Cf/Cc = 0.01.

(S400; C20/25) Cs/Cc = 30 Cf/Cc Gain (%) 0.01 14 0.02 14 0.03 13 0.04 13 0.05 13 0.06 12 0.07 12 0.08 12 0.09 12 0.1 12

Table 9. Variation of relative gain in percentage (%) versus unit cost ratio Cf/Cc for Cs/Cc = 30.

The optimal solutions obtained using the particular conditions imposed are shown in Table 7. From the above results, it is clearly seen that a significant cost saving between 09 and 15% can

The relative gains can be determined for various values of the unit cost ratios: Cs/Cc = 10; 20;

limited authorized template.

134 Optimum Composite Structures

4.2.3. Sensitivity analysis

be obtained by using this parametric study.

30; 40; 50; 60; 70; 80; 90; 100 for a given unit cost ratio Cf/Cc = 0.01

The following important conclusions are drawn on the basis of this chapter:


• The optimal solutions are found to be insensitive to changes in the shear constraint. Shear constraint is not usually critical in the optimal design of reinforced concrete T-beams under bending and thus can be excluded from problem formulation.

η Design strength factor

λ Compressive zone depth factor

γ<sup>s</sup> Partial safety factor for steel

Es Young's elastic modulus of steel Ecm Modulus of elasticity of concrete

μlimit Limit value of reduced moment

VRd,max Maximum resistant shear force

MRd, max Maximum resisting moment MEd Ultimate bending moment

VEd Ultimate shear force

pmin Minimum steel percentage pmax Maximum steel percentage

εyd Elastic limit strain

L Beam span

pressive concrete

design stress–strain relation

fyd Design yield strength of steel reinforcement

fyk Characteristic elastic limit for steel reinforcement

αlimit Limit value of relative depth of compressive concrete zone

w The total distribution load (dead load+ live load)

VG Maximum design shears under dead loads VQ Maximum design shears under live loads

MG Maximum design moments under dead loads MQ Maximum design moments under live loads

Fc Resultant compressive internal force for HSC

n Ratio of the modulus of elasticity of steel to that of concrete

Fs Resultant tensile internal force for steel

b Effective width of compressive flange

εc3 Strain at the maximum stress for the rectangular stress distribution com-

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εcu3 Ultimate strain for the rectangular stress distribution compressive concrete

