3. Calculation of bending reinforced concrete elements of a rectangular cross section

Consider the definition of the calculated resistance of reinforced concrete for bending reinforced concrete elements with single reinforcement. In order to show the universality of this method, regardless of the calculation method (force or deformation), first consider the term of the calculated resistance of the reinforced concrete for the force model laid down in the design standards SNiP 2.03.01-84\* [5].

For the stress-strain state shown in Figure 1, the equilibrium equation is written when ξ ≤ ξR, taking the sum of the moments relative to the neutral line:

$$f\_{yd}A\_s - f\_{cd}b\mathbf{x} = \mathbf{0}.\tag{4}$$

For the formula to take a familiar form, which is used in the resistance of materials in the calculations of metal, wooden, and stone structures, the left and right sides are multiplied by 6

Figure 1. Scheme of forces in the cross section of bending reinforced concrete element for single reinforcement.

In this formula Wc is the elastic moment of the resistance of the working cross section of concrete; 6D<sup>1</sup> is nothing more than the calculated resistance of the reinforced concrete to the

> <sup>f</sup> <sup>z</sup>М, <sup>1</sup> <sup>¼</sup> <sup>М</sup>Ed Wc

<sup>α</sup>Rf <sup>c</sup> <sup>¼</sup> MEd

<sup>f</sup> <sup>z</sup>М, <sup>2</sup> <sup>¼</sup> <sup>М</sup>Ed Wc

<sup>f</sup> <sup>z</sup>М,SNiP <sup>¼</sup> min <sup>6</sup><sup>f</sup> ydr<sup>f</sup> � <sup>6</sup><sup>f</sup>

8 ><

>:

6αRf <sup>c</sup>

The conditions are used (2) for the expression of the total calculated resistance:

bd<sup>2</sup> or <sup>6</sup>D<sup>1</sup> <sup>¼</sup> <sup>М</sup>Ed

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their…

Wc

, finally receiving

2 ydr<sup>2</sup> f 2f c

:

: (7)

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17

: (8)

bd<sup>2</sup> : (9)

: (10)

(11)

<sup>6</sup>D<sup>1</sup> <sup>¼</sup> <sup>6</sup>МEd

and written like

bend fzМ,1, namely:

Similarly, it is obtained with ξ > ξR:

The left part is denoted by D2 and then D2 = MEd/bd2

$$f\_{yd}A\_s(d-\infty) + f\_{cd}b\frac{\chi^2}{2} = M\_{Ed}.\tag{5}$$

The value of x is determined from Eq. (4) and substituted by expression (5). As a result of simple transformations received:

$$f\_{yd}\rho\_f - \frac{f\_{yd}^2 \rho\_f^2}{2f\_c} = \frac{M\_{Ed}}{bd^2}.\tag{6}$$

In formula (6) the left part is denoted by D1; then D<sup>1</sup> = MEd/bd2 . Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their… http://dx.doi.org/10.5772/intechopen.75122 17

Figure 1. Scheme of forces in the cross section of bending reinforced concrete element for single reinforcement.

For the formula to take a familiar form, which is used in the resistance of materials in the calculations of metal, wooden, and stone structures, the left and right sides are multiplied by 6 and written like

$$
\delta D\_1 = \frac{\theta M\_{Ed}}{bd^2} \quad \text{or} \quad \delta D\_1 = \frac{M\_{Ed}}{W\_\varepsilon}.\tag{7}
$$

In this formula Wc is the elastic moment of the resistance of the working cross section of concrete; 6D<sup>1</sup> is nothing more than the calculated resistance of the reinforced concrete to the bend fzМ,1, namely:

$$f\_{zM,1} = \frac{M\_{Ed}}{W\_c}.\tag{8}$$

Similarly, it is obtained with ξ > ξR:

4. By conducting preliminary calculations, the main calculated parameters are tabulated.

1. The only methodology for calculating composite materials for nonlinear deformation of

5. When obtaining new knowledge about the features of deformation of composite materials, it is enough to specify the value of the calculated resistance, and the method of calculation will remain unchanged, which greatly simplify the process of balance of norms [4].

6. Conducting comparative and estimating calculations of cross sections from different

Regarding the disadvantages, then they primarily relate suitability of this method for some

3. Calculation of bending reinforced concrete elements of a rectangular

Consider the definition of the calculated resistance of reinforced concrete for bending reinforced concrete elements with single reinforcement. In order to show the universality of this method, regardless of the calculation method (force or deformation), first consider the term of the calculated resistance of the reinforced concrete for the force model laid down in the

For the stress-strain state shown in Figure 1, the equilibrium equation is written when ξ ≤ ξR,

The value of x is determined from Eq. (4) and substituted by expression (5). As a result of

2 ydr<sup>2</sup> f 2f c

x2

<sup>¼</sup> MEd

f ydAsð Þþ d � x f cdb

<sup>f</sup> ydr<sup>f</sup> � <sup>f</sup>

f ydAs � f cdbx ¼ 0: (4)

.

<sup>2</sup> <sup>¼</sup> <sup>М</sup>Ed: (5)

bd<sup>2</sup> : (6)

The advantages of this method should include:

materials.

16 Cement Based Materials

cross section

design standards SNiP 2.03.01-84\* [5].

simple transformations received:

taking the sum of the moments relative to the neutral line:

In formula (6) the left part is denoted by D1; then D<sup>1</sup> = MEd/bd2

materials with classical material resistance.

2. Simplicity and convenience of the calculating process.

3. Ability to use different diagrams of deformation of materials.

4. Setting parameters of a stress-strain state at different load levels.

classes of materials, and some discomfort associated with using tables.

$$
\alpha\_{\mathbb{R}} f\_c = \frac{M\_{Ed}}{bd^2}.\tag{9}
$$

The left part is denoted by D2 and then D2 = MEd/bd2 , finally receiving

$$f\_{zM,2} = \frac{M\_{Ed}}{W\_c}.\tag{10}$$

The conditions are used (2) for the expression of the total calculated resistance:

$$f\_{zM,SNiP} = \min\left\{\begin{matrix} 6f\_{yd}\rho\_f - \frac{6f\_{yd}^2\rho\_f^2}{2f\_c} \\ 6\alpha\_{\mathbb{R}}f\_c \end{matrix} .\right. \tag{11}$$

Obtained by this way, calculated resistance of reinforced concrete for reinforcement А-400 and А-500 is shown in Table 1.

Similar expressions are obtained for nonlinear calculations. Write the value of the corresponding calculated resistances for different cases of destruction:

$$f\_{zM,1d\text{m}} = 6 \left( \frac{\int\_{\varepsilon}^{\varepsilon\_{\varepsilon}} \sigma\_{\varepsilon} \varepsilon\_{\varepsilon} d\varepsilon\_{\varepsilon}}{\left(\frac{\varepsilon\_{\varepsilon}}{\int\_{0}^{\varepsilon\_{\varepsilon}} \sigma\_{\varepsilon} d\varepsilon\_{\varepsilon}}\right)^{2} - \frac{\varepsilon\_{\varepsilon}}{\int\_{0}^{\varepsilon\_{\varepsilon}} \sigma\_{\varepsilon} d\varepsilon\_{\varepsilon}}} \right) \rho\_{f}^{2} f\_{yd}^{2} + 6 \rho\_{f} f\_{yd} \tag{12}$$

$$f\_{zM,2dm} = 6 \left( \frac{\int\_0^{\varepsilon\_\varepsilon} \sigma\_\varepsilon \varepsilon\_\varepsilon d\varepsilon\_\varepsilon}{\left(\frac{\varepsilon\_\varepsilon}{\int\_0^{\varepsilon\_\varepsilon} \sigma\_\varepsilon d\varepsilon\_\varepsilon}\right)^2 - \frac{\varepsilon\_\varepsilon}{\int\_0^{\varepsilon\_\varepsilon} \sigma\_\varepsilon d\varepsilon\_\varepsilon}} \right) \rho\_f^2 E\_s^2 \varepsilon\_\varepsilon^2 + 6\rho\_f E\_s \varepsilon\_\varepsilon. \tag{13}$$

To determine the corresponding calculated resistance, it is necessary for expressions (12) and (13) to apply an extreme criterion in the form:

$$\frac{d f\_{zM, idm}}{d \varepsilon\_{\varepsilon}} = 0, \quad \varepsilon\_{\varepsilon} \cdot \varepsilon[\varepsilon\_{cl, \varepsilon} \varepsilon\_{cu}].\tag{14}$$

f <sup>z</sup>М2,1dm ¼ 6

Note: Intermediate values are determined by straight-line interpolation.

Class of concrete Percentage of reinforcement r<sup>f</sup>

fyd = 375 MPa (A400С)

fyd = 450 MPa (A500С)

f <sup>z</sup>М2,2dm ¼ 6

Ð εc 0 σcεcdε ε2 c

Table 1. Calculated resistance of reinforced concrete to bend for single reinforcement fzМ,SNiP, MPa.

Ð εc 0 σcεсdε ε2 с

<sup>þ</sup> <sup>r</sup>fcf yc <sup>k</sup> � nk<sup>2</sup> � � <sup>þ</sup> <sup>ε</sup>cð Þ <sup>k</sup> � <sup>1</sup> <sup>2</sup>

0.05 0.50 1.00 1.25 1.50 1.75 2.00 2.50 3.00

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19

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their…

С8/10 1.11 9.49 15.47 15.66 15.66 15.66 15.66 15.66 15.66 С12/15 1.11 10.01 17.54 20.37 21.73 21.73 21.73 21.73 21.73 С16/20 1.12 10.33 18.83 22.39 25.50 28.14 28.61 28.61 28.61 С20/25 1.12 10.52 19.59 23.58 27.20 30.46 33.36 35.06 35.06 С25/30 1.12 10.63 20.02 24.25 28.17 31.78 35.07 40.08 40.08 С30/35 1.12 10.71 20.34 24.74 28.88 32.75 36.35 42.73 44.77 С32/40 1.12 10.77 20.58 25.13 29.44 33.50 37.33 44.26 49.13 С35/45 1.12 10.83 20.81 25.49 29.95 34.21 38.25 45.70 52.31 С40/50 1.12 10.87 20.97 25.73 30.30 34.68 38.86 46.66 53.69 С45/55 1.12 10.90 21.09 25.93 30.59 35.07 39.38 47.46 54.84 С50/60 1.12 10.93 21.22 26.13 30.87 35.46 39.89 48.26 55.99

С8/10 1.32 10.97 15.33 15.33 15.33 15.33 15.33 15.33 15.33 С12/15 1.33 11.71 19.85 21.22 21.22 21.22 21.22 21.22 21.22 С16/20 1.34 12.18 21.72 25.50 27.87 27.87 27.87 27.87 27.87 С20/25 1.34 12.45 22.81 27.20 31.07 34.06 34.06 34.06 34.06 С25/30 1.34 12.61 23.43 28.17 32.46 36.31 38.85 38.85 38.85 С30/35 1.34 12.72 23.88 28.88 33.49 37.71 41.54 43.31 43.31 С32/40 1.34 12.81 24.24 29.44 34.29 38.79 42.95 47.42 47.42 С35/45 1.34 12.89 24.57 29.95 35.03 39.81 44.28 51.91 51.91 С40/50 1.34 12.95 24.79 30.30 35.53 40.48 45.16 53.69 55.27 С45/55 1.34 12.99 24.98 30.59 35.94 41.05 45.90 54.84 58.28 С50/60 1.35 13.04 25.16 30.87 36.36 41.61 46.64 55.99 61.44

<sup>þ</sup> <sup>r</sup>fcf yc <sup>k</sup> � nk<sup>2</sup> � � <sup>þ</sup> <sup>r</sup><sup>f</sup> <sup>f</sup> yd <sup>k</sup>

k

kr<sup>f</sup> Е<sup>s</sup> <sup>k</sup><sup>2</sup> : (16)

<sup>2</sup> � <sup>k</sup> � �

<sup>2</sup> : (17)

The total calculated resistance to bend in calculating by the deformation model will be determined by the condition.

$$f\_{zM,dm} = \min\left\{ \begin{aligned} f\_{zM,1\mathcal{LM}} \frac{df\_{zM,1dm}}{d\varepsilon\_c} = 0, &\ \varepsilon\_c \ \epsilon[\varepsilon\_{cl,}\varepsilon\_{cu}];\\ f\_{zM,2\mathcal{LM}} \frac{df\_{zM,2dm}}{d\varepsilon\_c} = 0, &\ \varepsilon\_c \ \epsilon[\varepsilon\_{cl,}\varepsilon\_{cu}].\end{aligned} \tag{15}$$

For the further use of the expression (15), it is necessary to adopt a concrete deformation diagram. Adopted function of the deformation diagram does not have essential value, but it must satisfy the conditions for deformation of concrete. Accepting for deformation diagram for concrete Eurocode-2 [3], expression (3.14), the tabulation is executed so that the maximum fault in interpolation will not be more than 5%. The value of the calculated resistance to bending for single reinforcement for all classes of concrete and reinforcement classes А-400 and А-500 are shown in Table 2.

Similarly, the calculated resistance for bend for double reinforcement is obtained. For this purpose, the calculated resistance for different conditions of destruction of bending reinforced concrete elements for double reinforcement are determined:



Note: Intermediate values are determined by straight-line interpolation.

Obtained by this way, calculated resistance of reinforced concrete for reinforcement А-400 and

Similar expressions are obtained for nonlinear calculations. Write the value of the corresponding

Ð εc 0 σcdε<sup>c</sup>

Ð εc 0 σcdε<sup>c</sup>

To determine the corresponding calculated resistance, it is necessary for expressions (12) and

The total calculated resistance to bend in calculating by the deformation model will be deter-

df <sup>z</sup>М,1dm dε<sup>c</sup>

df <sup>z</sup>М,2dm dε<sup>c</sup>

For the further use of the expression (15), it is necessary to adopt a concrete deformation diagram. Adopted function of the deformation diagram does not have essential value, but it must satisfy the conditions for deformation of concrete. Accepting for deformation diagram for concrete Eurocode-2 [3], expression (3.14), the tabulation is executed so that the maximum fault in interpolation will not be more than 5%. The value of the calculated resistance to bending for single reinforcement for all classes of concrete and reinforcement classes А-400

Similarly, the calculated resistance for bend for double reinforcement is obtained. For this purpose, the calculated resistance for different conditions of destruction of bending reinforced

1

CCCCCA rf 2f 2

1

CCCCCA r2 f E2 s ε2

yd þ 6r<sup>f</sup> f yd, (12)

<sup>c</sup> þ 6r<sup>f</sup> Esεc: (13)

(15)

¼ 0, ε<sup>c</sup> є½ � εcl, εс<sup>u</sup> : (14)

¼ 0, ε<sup>c</sup> є½ � εcl, εс<sup>u</sup> ;

¼ 0, ε<sup>c</sup> є½ � εcl, εс<sup>u</sup> :

А-500 is shown in Table 1.

18 Cement Based Materials

calculated resistances for different cases of destruction:

f <sup>z</sup>М, <sup>1</sup>dm ¼ 6

f <sup>z</sup>М,2dm ¼ 6

(13) to apply an extreme criterion in the form:

f <sup>z</sup>М,dm ¼ min

concrete elements for double reinforcement are determined:

mined by the condition.

and А-500 are shown in Table 2.

Ð εc 0

0

BBBBB@

0

BBBBB@

Ð εc 0 σcdε<sup>c</sup>

Ð εc 0

> Ð εc 0 σcdε<sup>c</sup>

df <sup>z</sup>М,idm dε<sup>c</sup>

f <sup>z</sup>М, <sup>1</sup>ДМ,

8 >><

>>:

f <sup>z</sup>М, <sup>2</sup>ДМ,

σcεcdε<sup>c</sup>

σcεcdε<sup>c</sup>

!<sup>2</sup> � <sup>ε</sup><sup>c</sup>

!<sup>2</sup> � <sup>ε</sup><sup>c</sup>

Table 1. Calculated resistance of reinforced concrete to bend for single reinforcement fzМ,SNiP, MPa.

$$f\_{zM2,1d\text{mt}} = 6\frac{\int\_{\frac{0}{\varepsilon\_c^2}}^{\varepsilon\_c} + \rho\_f f\_{yc} \left(k - nk^2\right) + \varepsilon\_c \left(k - 1\right)^2 k\rho\_f E\_s}{k^2}.\tag{16}$$

$$f\_{zM2,2dm} = 6\frac{\int\_0^{\varepsilon\_c} \varepsilon\_c \, d\varepsilon}{k\_c^2} + \rho\_{\rm f\varepsilon} f\_{yc} \left(k - nk^2\right) + \rho\_{\rm f} f\_{yd} \left(k^2 - k\right)}.\tag{17}$$


In the given expressions, k is determined from the first equation of equilibrium under the

The total calculated resistance to bend in double reinforcement according to the deformation

df <sup>z</sup>М,1dm dε<sup>c</sup>

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their…

df <sup>z</sup>М,2dm dε<sup>c</sup>

df <sup>z</sup>М,3dm dε<sup>c</sup>

df <sup>z</sup>М,4dm dε<sup>c</sup>

In Table 3 the expression of the calculated resistance to bend for double reinforcement is derived taking the diagram of deformation of concrete in the form of the function Eurocode-2 [3].

As can be seen from Tables 2 and 3 in some cases, double reinforcement significantly (more than three times) increases the calculated resistance of reinforced concrete of bending elements and accordingly increases their bearing capacity. In this way, the reinforcement can greatly enhance the compressed area of concrete of bending reinforced concrete elements. For comparison, the data of the calculated resistance to bend for double reinforcement are presented by

Compare the calculated resistance of reinforced concrete to bending defined by the force model and deformation method for single and double reinforcement. As can be seen from Tables 5 and 6, the calculated resistance of reinforced concrete to the bend differs within the limits of the calculated fault. This makes it possible to say that for heavy concrete classes С8/ 10÷С50/60 and ordinary reinforcement classes А-400 and А-500, calculations of the strength of the cross sections of bending reinforced concrete elements with single and double reinforcement can be performed on any of the mentioned methods. By these ways, the maximum

One of the main advantages of the deformation model in comparison with the force one is the possibility of obtaining the parameters of the stress-strain state for the operational load. Let's show how this can be done using the method of calculated resistance of reinforced concrete. For this purpose, the tensions in the bending reinforced concrete element are determined σzМ,ДМ under operational loads, at which cross sections of the element can work without cracks at М < МW, with cracks in the stretched zone at М ≥ МW, without cracks at М ≥ М<sup>W</sup> (areas in the block between the cracks). In this case, it is proposed to determine the tension in

þ Esrfcð Þ 1 � nkW

k2 W 2

kW <sup>þ</sup> Esrfð Þ kW � <sup>1</sup> <sup>2</sup>

kW

: (21)

¼ 0, ε<sup>c</sup> є½ � εcl, εс<sup>u</sup> ;

http://dx.doi.org/10.5772/intechopen.75122

¼ 0, ε<sup>c</sup> є½ � εcl, εс<sup>u</sup> ;

(20)

21

¼ 0, ε<sup>c</sup> є½ � εcl, εс<sup>u</sup> ;

¼ 0, ε<sup>c</sup> є½ � εcl, εс<sup>u</sup> :

f <sup>z</sup>М2, <sup>1</sup>dm,

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

f <sup>z</sup>М2, <sup>2</sup>dm,

f <sup>z</sup>М2, <sup>3</sup>dm,

f <sup>z</sup>М2, <sup>4</sup>dm,

conditions of the destruction of the element.

model will be determined by the condition:

f <sup>z</sup>М2, dm ¼ min

methodology of SNiP 2.03.01–84\* [5] (Table 4).

difference will be within 8% and only for certain conditions.

the reinforced concrete for cross sections until formation of cracks.

þ Ð εсtu 0 σсtεctdε ε3 c,W

Ð <sup>ε</sup>c,W 0 σcεсdε ε3 c,W

σWz<sup>М</sup> ¼ 6εc,W �

Note: Intermediate values are determined by straight-line interpolation.

Table 2. Calculated resistance of reinforced concrete to bend for single reinforcement fzМ,dm, MPa.

$$f\_{zM2,3dm} = 6\varepsilon\_c \frac{\int\_0^{\varepsilon\_c} \rho\_c d\varepsilon}{k^3} + E\_s \rho\_{fc} (1 - nk)^2 k + E\_s \rho\_f (k - 1)^2 k}. \tag{18}$$

$$f\_{zM2,4dm} = \epsilon \varepsilon\_c \frac{\int\_0^{\varepsilon\_c} \varepsilon\_c d\epsilon}{k\_c^3} + E\_s \rho\_f (1 - nk)^2 k + \frac{f\_{yl}}{\varepsilon\_c} \rho\_f (k - 1) k}{k^2}.\tag{19}$$

In the given expressions, k is determined from the first equation of equilibrium under the conditions of the destruction of the element.

The total calculated resistance to bend in double reinforcement according to the deformation model will be determined by the condition:

$$f\_{zM2,dm} = \min \begin{cases} f\_{zM2,1dm'} \frac{df\_{zM,1dm}}{d\varepsilon\_c} = 0, & \varepsilon\_c \in [\varepsilon\_{cl,}\varepsilon\_{cu}]; \\ f\_{zM2,2dm'} \frac{df\_{zM,2dm}}{d\varepsilon\_c} = 0, & \varepsilon\_c \in [\varepsilon\_{cl,}\varepsilon\_{cu}]; \\ f\_{zM2,3dm'} \frac{df\_{zM,3dm}}{d\varepsilon\_c} = 0, & \varepsilon\_c \in [\varepsilon\_{cl,}\varepsilon\_{cu}]; \\ f\_{zM2,4dm'} \frac{df\_{zM,4dm}}{d\varepsilon\_c} = 0, & \varepsilon\_c \in [\varepsilon\_{cl,}\varepsilon\_{cu}]. \end{cases} \tag{20}$$

In Table 3 the expression of the calculated resistance to bend for double reinforcement is derived taking the diagram of deformation of concrete in the form of the function Eurocode-2 [3].

As can be seen from Tables 2 and 3 in some cases, double reinforcement significantly (more than three times) increases the calculated resistance of reinforced concrete of bending elements and accordingly increases their bearing capacity. In this way, the reinforcement can greatly enhance the compressed area of concrete of bending reinforced concrete elements. For comparison, the data of the calculated resistance to bend for double reinforcement are presented by methodology of SNiP 2.03.01–84\* [5] (Table 4).

Compare the calculated resistance of reinforced concrete to bending defined by the force model and deformation method for single and double reinforcement. As can be seen from Tables 5 and 6, the calculated resistance of reinforced concrete to the bend differs within the limits of the calculated fault. This makes it possible to say that for heavy concrete classes С8/ 10÷С50/60 and ordinary reinforcement classes А-400 and А-500, calculations of the strength of the cross sections of bending reinforced concrete elements with single and double reinforcement can be performed on any of the mentioned methods. By these ways, the maximum difference will be within 8% and only for certain conditions.

One of the main advantages of the deformation model in comparison with the force one is the possibility of obtaining the parameters of the stress-strain state for the operational load. Let's show how this can be done using the method of calculated resistance of reinforced concrete. For this purpose, the tensions in the bending reinforced concrete element are determined σzМ,ДМ under operational loads, at which cross sections of the element can work without cracks at М < МW, with cracks in the stretched zone at М ≥ МW, without cracks at М ≥ М<sup>W</sup> (areas in the block between the cracks). In this case, it is proposed to determine the tension in the reinforced concrete for cross sections until formation of cracks.

f <sup>z</sup>М2,3dm ¼ 6ε<sup>c</sup>

Note: Intermediate values are determined by straight-line interpolation.

Class of concrete Percentage of reinforcement r<sup>f</sup>

20 Cement Based Materials

fyd = 375 MPa (A400С)

fyd = 450 MPa (A500С)

f <sup>z</sup>М2,4dm ¼ 6ε<sup>c</sup>

Ð εc 0 σcεсdε ε3 с

Table 2. Calculated resistance of reinforced concrete to bend for single reinforcement fzМ,dm, MPa.

Ð εc 0 σcεсdε ε3 с

<sup>þ</sup> Esrfcð Þ <sup>1</sup> � nk <sup>2</sup>

<sup>þ</sup> Esr<sup>f</sup>сð Þ <sup>1</sup> � nk <sup>2</sup>

k

0.05 0.50 1.00 1.25 1.50 1.75 2.00 2.50 3.00

С8/10 1.10 9.44 14.68 15.12 15.43 15.67 15.86 16.13 16.32 С12/15 1.11 9.97 17.38 20.09 20.85 21.27 21.60 22.10 22.45 С16/20 1.11 10.30 18.70 22.19 25.20 27.38 27.90 28.71 29.29 С20/25 1.11 10.49 19.48 23.40 26.95 30.11 32.88 34.82 35.65 С25/30 1.11 10.60 19.91 24.08 27.93 31.46 34.66 39.64 40.69 С30/35 1.12 10.68 20.24 24.59 28.66 32.45 35.96 42.12 45.45 С32/40 1.12 10.75 20.49 24.98 29.23 33.22 36.96 43.69 49.26 С35/45 1.12 10.81 20.72 25.35 29.76 33.94 37.90 45.16 51.52 С40/50 1.12 10.84 20.88 25.60 30.11 34.42 38.53 46.14 52.94 С45/55 1.12 10.87 21.01 25.80 30.40 34.82 39.05 46.95 54.10 С50/60 1.12 10.90 21.14 26.00 30.69 35.21 39.56 47.75 55.26

С8/10 1.32 10.90 14.57 15.02 15.35 15.60 15.79 16.07 16.27 С12/15 1.33 11.66 19.40 20.15 20.70 21.13 21.48 21.99 22.35 С16/20 1.33 12.13 21.53 25.17 26.50 27.16 27.71 28.53 29.13 С20/25 1.33 12.41 22.65 26.95 30.69 32.67 33.42 34.57 35.43 С25/30 1.34 12.57 23.28 27.93 32.12 35.82 37.89 39.33 40.40 С30/35 1.34 12.69 23.74 28.66 33.18 37.28 40.93 43.80 45.10 С32/40 1.34 12.78 24.11 29.23 33.99 38.39 42.40 48.01 49.53 С35/45 1.34 12.86 24.44 29.76 34.75 39.42 43.78 51.43 54.52 С40/50 1.34 12.92 24.67 30.11 35.26 40.11 44.68 52.94 58.52 С45/55 1.34 12.96 24.86 30.40 35.68 40.69 45.43 54.10 61.40 С50/60 1.34 13.01 25.04 30.69 36.10 41.25 46.17 55.26 63.37

<sup>k</sup> <sup>þ</sup> Esrfð Þ <sup>k</sup> � <sup>1</sup> <sup>2</sup>

<sup>ε</sup><sup>c</sup> rfð Þ k � 1 k

<sup>k</sup> <sup>þ</sup> <sup>f</sup> yd

k <sup>k</sup><sup>2</sup> : (18)

<sup>2</sup> : (19)

$$
\sigma\_{\mathsf{W}\succeq\mathsf{M}} = \mathsf{f}\varepsilon\_{\mathsf{c},\mathsf{W}} \times \frac{\int\_{0}^{\varepsilon\_{\mathsf{c}}\rho\_{\mathsf{c}}d\varepsilon} + \int\_{0}^{\varepsilon\_{\mathsf{c}\mathsf{c}}} \rho\_{\mathsf{c}t}e\_{\mathsf{c}t}d\varepsilon}{\int\_{\mathsf{c}\_{\mathsf{c}}\rho\_{\mathsf{c}}}^{3}} + E\_{\mathsf{s}}\rho\_{\mathsf{f}\complement}(1-n\mathsf{k}\_{\mathsf{W}})^{2}\mathsf{k}\_{\mathsf{W}} + E\_{\mathsf{s}}\rho\_{\mathsf{f}}(\mathsf{k}\_{\mathsf{W}}-1)^{2}\mathsf{k}\_{\mathsf{W}}}.\tag{21}
$$


Table 3. Calculated resistance of reinforced concrete to bend for double reinforcement fzМ2,dm, MPa.

It is noted that the tension σWz<sup>М</sup> also allows to determine the moment of formation of cracks, so depending on the tasks, it can also be called the calculated resistance of the reinforced concrete to the bend until formation of cracks.

The tensions between the cracks are determined by

Note: Intermediate values are determined by straight-line interpolation.

σcεdε

þ 6

All of the above tensions in concrete are determined by expression:

ð Þ <sup>k</sup> � <sup>1</sup> <sup>2</sup> <sup>Ð</sup>

Table 4. Calculated resistance of reinforced concrete to bend for double reinforcement fzМ2,SNiP, MPa.

Class of concrete r<sup>f</sup> = 0.01 r<sup>f</sup> = 0.02 r<sup>f</sup> = 0.03

0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75

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23

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their…

С8/10 18.20 20.06 21.04 26.23 36.61 41.20 31.51 47.37 60.48 С12/15 19.37 20.58 21.17 32.29 38.67 41.71 37.58 53.43 61.64 С16/20 20.10 20.90 21.25 36.06 39.97 42.04 44.47 57.20 62.37 С20/25 20.52 21.09 21.30 37.77 40.73 42.23 50.91 58.91 62.80 С25/30 20.76 21.20 21.32 38.73 41.15 42.33 53.91 59.87 63.04 С30/35 20.94 21.28 21.34 39.45 41.47 42.41 55.52 60.59 63.21 С32/40 21.08 21.34 21.36 40.00 41.72 42.48 56.77 61.14 63.35 С35/45 21.21 21.40 21.37 40.52 41.95 42.53 57.93 61.66 63.48 С40/50 21.30 21.43 21.38 40.87 42.10 42.57 58.71 62.00 63.57 С45/55 21.37 21.47 21.39 41.15 42.23 42.60 59.36 62.29 63.64 С50/60 21.44 21.50 21.40 41.44 42.36 42.63 60.01 62.58 63.71

С8/10 20.51 23.15 24.78 26.60 37.87 47.32 32.24 49.15 66.05 С12/15 22.31 24.07 25.11 32.49 43.72 48.64 38.12 55.03 70.58 С16/20 23.44 24.64 25.32 39.14 46.02 49.47 44.78 61.69 72.44 С20/25 24.10 24.98 25.44 43.13 47.37 49.96 50.97 67.18 73.54 С25/30 24.47 25.17 25.51 44.63 48.13 50.23 55.76 68.89 74.15 С30/35 24.75 25.31 25.56 45.73 48.70 50.43 60.22 70.16 74.61 С32/40 24.97 25.42 25.60 46.59 49.14 50.59 64.33 71.15 74.97 С35/45 25.17 25.52 25.64 47.39 49.55 50.74 66.68 72.07 75.30 С40/50 25.30 25.59 25.66 47.93 49.82 50.84 67.88 72.68 75.52 С45/55 25.41 25.65 25.68 48.37 50.05 50.92 68.88 73.19 75.70 С50/60 25.52 25.70 25.70 48.82 50.27 51.00 69.89 73.70 75.89

rfc/r<sup>f</sup>

n = 0.06–0.1 fyd = 375 MPa (А400С)

n = 0.06–0.1 fyd = 450 MPa (А500С)

k 2 ε2 ctu

εсtu 0

σс<sup>t</sup>εсtdε

þ 6

k � 1

<sup>k</sup> <sup>σ</sup>s,mr<sup>f</sup> <sup>þ</sup> <sup>6</sup>

ð Þ <sup>1</sup> � kn <sup>2</sup> k kð Þ � <sup>1</sup> <sup>r</sup>fcσs,m:

(23)

<sup>σ</sup>s,m Esð Þ k�1 0

σmz<sup>М</sup> ¼ 6

E2

<sup>s</sup> ð Þ <sup>k</sup> � <sup>1</sup> <sup>2</sup> <sup>Ð</sup>

k2 σ2 s,m

Tension in a cross section with a crack in the stretched zone at М ≥ М<sup>W</sup> is determined by the

$$\sigma\_{2zM} = 6\frac{\int\_0^{\varepsilon\_{\varepsilon}} \sigma\_{\varepsilon} \varepsilon\_{\varepsilon} d\varepsilon}{k^2 \varepsilon\_{\varepsilon}^2} + \frac{\int\_0^{\varepsilon\_{\text{th}}} \sigma\_{\varepsilon t} \varepsilon\_{\text{tf}} d\varepsilon}{k^2 \varepsilon\_{\varepsilon}^2} + 6\varepsilon\_{\varepsilon} E\_s \rho\_f \frac{\left(1 - nk\right)^2}{k} + 6\varepsilon\_{\varepsilon} E\_s \rho\_f \frac{\left(k - 1\right)^2}{k} + \frac{\left(k - 1\right)}{k} \Delta \sigma\_{\text{s},x} \rho\_f. \tag{22}$$



Note: Intermediate values are determined by straight-line interpolation.

Table 4. Calculated resistance of reinforced concrete to bend for double reinforcement fzМ2,SNiP, MPa.

The tensions between the cracks are determined by

It is noted that the tension σWz<sup>М</sup> also allows to determine the moment of formation of cracks, so depending on the tasks, it can also be called the calculated resistance of the reinforced

Tension in a cross section with a crack in the stretched zone at М ≥ М<sup>W</sup> is determined by the

ð Þ <sup>1</sup> � nk <sup>2</sup>

<sup>k</sup> <sup>þ</sup> <sup>6</sup>εcEsr<sup>f</sup>

ð Þ <sup>k</sup> � <sup>1</sup> <sup>2</sup>

<sup>k</sup> <sup>þ</sup> ð Þ <sup>k</sup> � <sup>1</sup>

<sup>k</sup> <sup>Δ</sup>σs,хr<sup>f</sup> : (22)

concrete to the bend until formation of cracks.

Ð εсtu 0

> k 2 ε2 c

σс<sup>t</sup>εctdε

Note: Intermediate values are determined by straight-line interpolation.

þ 6εcEsrfc

Table 3. Calculated resistance of reinforced concrete to bend for double reinforcement fzМ2,dm, MPa.

Class of concrete r<sup>f</sup> = 0.01 r<sup>f</sup> = 0.02 r<sup>f</sup> = 0.03

0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75

С8/10 18.09 20.02 21.04 25.81 35.75 41.17 31.54 46.74 60.39 С12/15 19.28 20.55 21.16 31.33 38.53 41.70 37.42 52.38 61.58 С16/20 20.03 20.88 21.19 35.78 39.85 42.03 43.99 56.93 62.33 С20/25 20.46 21.07 21.23 37.52 40.63 42.22 50.11 58.67 62.76 С25/30 20.71 21.17 21.26 38.51 41.06 42.32 53.39 59.66 63.01 С30/35 20.89 21.22 21.29 39.24 41.39 42.35 55.04 60.39 63.19 С32/40 21.02 21.26 21.32 39.80 41.64 42.38 56.31 60.95 63.33 С35/45 21.14 21.31 21.36 40.33 41.87 42.41 57.50 61.48 63.44 С40/50 21.21 21.35 21.39 40.68 42.03 42.44 58.30 61.84 63.50 С45/55 21.28 21.39 21.43 40.98 42.14 42.46 58.95 62.13 63.53 С50/60 21.35 21.43 21.46 41.27 42.25 42.50 59.61 62.42 63.55

С8/10 20.23 23.59 25.13 27.70 39.59 48.97 34.49 52.69 70.91 С12/15 22.45 24.35 25.25 33.11 44.76 49.73 40.27 58.17 73.21 С16/20 23.52 24.82 25.31 39.06 46.91 50.20 46.71 64.31 74.28 С20/25 24.14 25.08 25.36 43.38 48.03 50.45 52.71 68.76 74.90 С25/30 24.49 25.18 25.41 44.81 48.65 50.50 57.47 70.19 75.25 С30/35 24.74 25.26 25.45 45.87 49.11 50.55 61.98 71.25 75.50 С32/40 24.92 25.33 25.49 46.67 49.46 50.60 65.05 72.05 75.68 С35/45 25.08 25.40 25.53 47.42 49.78 50.65 66.84 72.80 75.74 С40/50 25.18 25.46 25.56 47.92 49.98 50.69 67.99 73.29 75.78 С45/55 25.27 25.50 25.60 48.32 50.13 50.73 68.92 73.70 75.82 С50/60 25.35 25.55 25.64 48.70 50.24 50.77 69.87 74.08 75.87

rfc/r<sup>f</sup>

n = 0.06–0.1 fyd = 375 MPa (А400С)

22 Cement Based Materials

n = 0.06–0.1 fyd = 450 MPa (А500С)

σ2z<sup>М</sup> ¼ 6

Ð εc 0

k 2 ε2 c þ

σcεсdε

$$\sigma\_{mzM} = 6 \frac{E\_s^2 (k-1)^2 \int\_0^{\frac{\sigma\_{\rm tm}}{L\_\varepsilon(k-1)}} \sigma\_\varepsilon \varepsilon d\varepsilon}{k^2 \sigma\_{s,m}^2} + 6 \frac{(k-1)^2 \int\_0^{\varepsilon\_{\rm tm}} \sigma\_{\varepsilon t} \varepsilon\_{\rm t} d\varepsilon}{k^2 \varepsilon\_{\rm tm}^2} + 6 \frac{k-1}{k} \sigma\_{s,m} \rho\_f + 6 \frac{(1-kn)^2}{k(k-1)} \rho\_f \sigma\_{s,m} \tag{23}$$

All of the above tensions in concrete are determined by expression:


Table 5. Comparison of the calculated resistance of reinforced concrete to bend for single reinforcement fzМ,dm/fzМ,SNiP.

$$
\sigma\_{\rm izM} = \frac{M}{\mathcal{W}\_c}.\tag{24}
$$

The tension in the reinforcement in the cross section with the crack in the stretched zone are calculated at known values k, εc, Δσs,x. Average tensions in the reinforcement on the section in the block between the cracks σs,m are defined as the arithmetic mean of the tensions that are

Table 6. Comparison of the calculated resistance of reinforced concrete to bend for double reinforcement fzМ2,dm/fzМ2,SNiP.

The curvature of the cross sections of reinforced concrete elements, taking into account the

hypothesis of plane cross sections, is determined by the expression

Class of concrete r<sup>f</sup> = 0.01 r<sup>f</sup> = 0.02 r<sup>f</sup> = 0.03

1 2 3 4 5 6 7 8 9 10

С8/10 0.994 0.998 1.000 0.984 0.977 0.999 1.001 0.987 0.999 С12/15 0.996 0.998 1.000 0.970 0.996 1.000 0.996 0.980 0.999 С16/20 0.997 0.999 0.997 0.992 0.997 1.000 0.989 0.995 0.999 С20/25 0.997 0.999 0.997 0.993 0.998 1.000 0.984 0.996 0.999 С25/30 0.997 0.999 0.997 0.994 0.998 1.000 0.990 0.996 1.000 С30/35 0.998 0.997 0.998 0.995 0.998 0.999 0.991 0.997 1.000 С32/40 0.997 0.996 0.998 0.995 0.998 0.998 0.992 0.997 1.000 С35/45 0.997 0.996 0.999 0.995 0.998 0.997 0.993 0.997 0.999 С40/50 0.996 0.996 1.001 0.996 0.998 0.997 0.993 0.997 0.999 С45/55 0.996 0.996 1.002 0.996 0.998 0.997 0.993 0.997 0.998 С50/60 0.996 0.997 1.003 0.996 0.997 0.997 0.993 0.997 0.998

С8/10 0.986 1.019 1.014 1.041 1.045 1.035 1.070 1.072 1.074 С12/15 1.006 1.012 1.005 1.019 1.024 1.022 1.056 1.057 1.037 С16/20 1.003 1.007 0.999 0.998 1.019 1.015 1.043 1.042 1.025 С20/25 1.002 1.004 0.997 1.006 1.014 1.010 1.034 1.024 1.019 С25/30 1.000 1.000 0.996 1.004 1.011 1.005 1.031 1.019 1.015 С30/35 0.999 0.998 0.996 1.003 1.009 1.002 1.029 1.015 1.012 С32/40 0.998 0.996 0.995 1.002 1.007 1.000 1.011 1.013 1.010 С35/45 0.996 0.995 0.996 1.001 1.005 0.998 1.002 1.010 1.006 С40/50 0.995 0.995 0.996 1.000 1.003 0.997 1.002 1.008 1.004 С45/55 0.994 0.994 0.997 0.999 1.002 0.996 1.000 1.007 1.002 С50/60 0.993 0.994 0.997 0.998 0.999 0.995 1.000 1.005 1.000

0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75

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25

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their…

rfc/r<sup>f</sup>

n = 0.06–0.1 fyd = 375 MPa (А400С)

n = 0.06–0.1 fyd = 450 MPa (А500С)

determined by expressions (22) and (23).

Parameters of the stress-strain state at the operating load levels are necessary for determining the deflection and width of the crack opening. Therefore, the basic parameters that are necessary for this will be: tension in the reinforcement and curvature.

The tension in the reinforcement until formation of cracks is determined by expression

$$
\sigma\_{\text{s,W}} = (k\_W - 1)\varepsilon\_{\text{c,W}} \tag{25}
$$

under certain values kW, εc,W.


Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their… http://dx.doi.org/10.5772/intechopen.75122 25

Table 6. Comparison of the calculated resistance of reinforced concrete to bend for double reinforcement fzМ2,dm/fzМ2,SNiP.

<sup>σ</sup>iz<sup>М</sup> <sup>¼</sup> <sup>М</sup> Wc

Table 5. Comparison of the calculated resistance of reinforced concrete to bend for single reinforcement fzМ,dm/fzМ,SNiP.

0.05 0.50 1.00 1.25 1.50 1.75 2.00 2.50 3.00

С8/10 0.997 0.995 0.949 0.965 0.985 1.001 1.013 1.030 1.042 С12/15 0.997 0.996 0.991 0.986 0.960 0.979 0.994 1.017 1.033 С16/20 0.996 0.997 0.993 0.991 0.988 0.973 0.975 1.003 1.024 С20/25 0.996 0.997 0.994 0.992 0.991 0.989 0.986 0.993 1.017 С25/30 0.996 0.998 0.995 0.993 0.992 0.990 0.988 0.989 1.015 С30/35 0.996 0.998 0.995 0.994 0.992 0.991 0.989 0.986 1.015 С32/40 0.996 0.998 0.996 0.994 0.993 0.992 0.990 0.987 1.003 С35/45 0.996 0.998 0.996 0.995 0.994 0.992 0.991 0.988 0.985 С40/50 0.996 0.998 0.996 0.995 0.994 0.993 0.991 0.989 0.986 С45/55 0.996 0.998 0.996 0.995 0.994 0.993 0.992 0.989 0.986 С50/60 0.996 0.997 0.996 0.995 0.994 0.993 0.992 0.990 0.987

С8/10 0.998 0.993 0.951 0.980 1.001 1.018 1.030 1.049 1.061 С12/15 0.997 0.995 0.977 0.950 0.976 0.996 1.012 1.037 1.054 С16/20 0.997 0.996 0.991 0.987 0.951 0.975 0.994 1.024 1.045 С20/25 0.996 0.997 0.993 0.991 0.988 0.959 0.981 1.015 1.040 С25/30 0.996 0.997 0.994 0.992 0.990 0.987 0.975 1.012 1.040 С30/35 0.996 0.997 0.994 0.992 0.991 0.989 0.985 1.012 1.042 С32/40 0.996 0.997 0.995 0.993 0.991 0.990 0.987 1.012 1.044 С35/45 0.996 0.998 0.995 0.993 0.992 0.990 0.989 0.991 1.050 С40/50 0.996 0.998 0.995 0.994 0.992 0.991 0.989 0.986 1.059 С45/55 0.996 0.998 0.995 0.994 0.993 0.991 0.990 0.986 1.053 С50/60 0.996 0.998 0.995 0.994 0.993 0.991 0.990 0.987 1.031

Parameters of the stress-strain state at the operating load levels are necessary for determining the deflection and width of the crack opening. Therefore, the basic parameters that are neces-

The tension in the reinforcement until formation of cracks is determined by expression

sary for this will be: tension in the reinforcement and curvature.

under certain values kW, εc,W.

Class of concrete Percentage of reinforcement r<sup>f</sup>

24 Cement Based Materials

fyd = 375 MPa (A400С)

fyd = 450 MPa (A500С)

: (24)

σs,W ¼ ð Þ kW � 1 εс,W, (25)

The tension in the reinforcement in the cross section with the crack in the stretched zone are calculated at known values k, εc, Δσs,x. Average tensions in the reinforcement on the section in the block between the cracks σs,m are defined as the arithmetic mean of the tensions that are determined by expressions (22) and (23).

The curvature of the cross sections of reinforced concrete elements, taking into account the hypothesis of plane cross sections, is determined by the expression

$$|\cdot|\_{r} = \sum \varepsilon / d,\tag{26}$$

Calculation of deflections is performed in the following order:

solution of this problem.

capacity of the beam.

determined:

σ<sup>z</sup><sup>М</sup> ¼ Ме=Wc ! Σε ! 1=r ¼ Σε=d ! f : (33)

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27

A separate important issue in the theory of reinforced concrete is the consideration of regime loads and influences: long-term, quasi-constant, low cycle, temperature, humidity and others. Thus, taking into account the long-term load can be realized by introducing the creep coefficient to the curvature or by introducing into the calculation of the deformation diagrams with the corresponding parameters. The calculation of regime load under the first condition can be carried out according to the given method by using tables for short-term load. When performing calculations under the second condition, it is necessary to use the tables obtained for the corresponding parameters of the diagrams. Similar tables can be made for virtually all regime loads and influences, which greatly simplify the calculations of strength, crack resistance, stiffness and width of crack opening. This is an issue that needs to be studied in detail, but the use of the calculated resistance of reinforced concrete gives confidence in the successful

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their…

4. Examples of calculation of bending reinforced concrete elements

Example 1. Reinforced concrete beam with working cross section b � d = 20 � 45 sm is made of concrete of class С25/30 and reinforced 4∅25 of steel of class А500С. Determine the carrying

Solution. The percentage of beam reinforcement with stretched reinforcement is calculated:

According to the tables the calculated resistance of the reinforced concrete to the bend is

f <sup>z</sup><sup>М</sup> ¼ 37:12 MPa:

Example 2. Reinforced concrete beam with working cross section b � d = 30 � 45 sm is made of concrete of class С25/30 and steel of class А400С and should take an external moment

<sup>20</sup> � <sup>45</sup> <sup>¼</sup> <sup>2</sup>:181%:

<sup>6</sup> <sup>37</sup>:<sup>12</sup> � <sup>10</sup>�<sup>3</sup> <sup>¼</sup> <sup>250</sup>:56 kNm:

bd � <sup>100</sup>% <sup>¼</sup> <sup>19</sup>:<sup>63</sup>

<sup>6</sup> <sup>f</sup> <sup>z</sup><sup>М</sup> <sup>¼</sup> <sup>20</sup> � <sup>452</sup>

Solution. The moment of resistance of the concrete cross section is determined:

<sup>r</sup><sup>f</sup> <sup>¼</sup> <sup>А</sup><sup>s</sup>

The carrying capacity of the beam is calculated by the formula:

МЕ<sup>d</sup> <sup>¼</sup> Wcf <sup>z</sup><sup>М</sup> <sup>¼</sup> bd<sup>2</sup>

МEd = 266.46 kNm. Determine element reinforcement.

where Σε is the total deformation of fibrous concrete fibers and stretched reinforcement.

The total deformation of fibrous concrete fibers and stretched reinforcement must be determined by the following formulas:

• For cross sections without cracks at М < МW:

$$
\sum \varepsilon = \varepsilon\_{\mathfrak{c}, \mathcal{W}} + \varepsilon\_{\mathfrak{s}, \mathcal{W}} = \varepsilon\_{\mathfrak{c}, \mathcal{W}} + \sigma\_{\mathfrak{s}, \mathcal{W}} / E\_{\mathfrak{s}}.\tag{27}
$$

• For cross sections with a crack in the stretched zone:

$$
\sum \varepsilon = \varepsilon\_{c,2} + \varepsilon\_{s, \text{ffc}} = \varepsilon\_{c,2} + (k\_{W,2} - 1)\varepsilon\_{c,2}.\tag{28}
$$

• For cross sections without cracks at М ≥ М<sup>W</sup>

$$
\sum \varepsilon = \varepsilon\_{c,m} + \varepsilon\_{s,m} = \frac{\sigma\_{s,m}}{E\_s(k-1)} + \frac{\sigma\_{s,m}}{E\_s}.\tag{29}
$$

Deflections are determined by curvature by using numerical methods.

According to the given method, tables have been developed, which depending on the accepted parameters allow to determining the resistance of the concrete, the stresses in the reinforced concrete and reinforcement, and the total relative deformation of the cross section. For this purpose, the deformation diagram was adopted in the form of Eurocode-2 function [3]. These tables are given in [6].

The calculation of the strength of the cross sections of bending reinforced concrete elements and crack resistance is recommended to be performed according to the formula:

$$\frac{M\_{Ed}}{W\_c} \le f\_{zM}(f\_{\text{WzM}}) \,. \tag{30}$$

Calculation of tension limitation in the reinforcement is carried out as follows:

$$
\sigma\_{zM} = \frac{M\_{\varepsilon}}{W\_{\varepsilon}} \to \varepsilon\_{si} \to \sigma\_{s} \le \sigma\_{s, Tablle\prime} \tag{31}
$$

where σs,Table is the tension in the reinforcement, in which there is no need to determine the width of the cracks opening, which are determined by Table 2.5 [7].

The calculation of the width of the cracks opening is carried out in same scheme:

$$
\sigma\_{zM} = M\_e / W\_c \to \varepsilon\_{si} \to \sigma\_{si} \to \mathcal{S}\_r \to \mathcal{W}\_k. \tag{32}
$$

Calculation of deflections is performed in the following order:

<sup>1</sup>=<sup>r</sup> <sup>¼</sup> <sup>X</sup>ε=d, (26)

<sup>X</sup><sup>ε</sup> <sup>¼</sup> <sup>ε</sup>с,W <sup>þ</sup> <sup>ε</sup>s,W <sup>¼</sup> <sup>ε</sup>с,W <sup>þ</sup> <sup>σ</sup>s,W <sup>=</sup>Es: (27)

<sup>X</sup><sup>ε</sup> <sup>¼</sup> <sup>ε</sup>c, <sup>2</sup> <sup>þ</sup> <sup>ε</sup>s,fic <sup>¼</sup> <sup>ε</sup>c,<sup>2</sup> <sup>þ</sup> ð Þ kW, <sup>2</sup> � <sup>1</sup> <sup>ε</sup>c,2: (28)

σs,m Es

: (29)

� �: (30)

! ε<sup>s</sup><sup>і</sup> ! σ<sup>s</sup> ≤ σs,Table, (31)

σ<sup>z</sup><sup>М</sup> ¼ Ме=Wc ! ε<sup>s</sup><sup>і</sup> ! σ<sup>s</sup><sup>і</sup> ! Sr ! Wk: (32)

Esð Þ <sup>k</sup> � <sup>1</sup> <sup>þ</sup>

where Σε is the total deformation of fibrous concrete fibers and stretched reinforcement.

<sup>X</sup><sup>ε</sup> <sup>¼</sup> <sup>ε</sup>c,m <sup>þ</sup> <sup>ε</sup>s,m <sup>¼</sup> <sup>σ</sup>s,m

According to the given method, tables have been developed, which depending on the accepted parameters allow to determining the resistance of the concrete, the stresses in the reinforced concrete and reinforcement, and the total relative deformation of the cross section. For this purpose, the deformation diagram was adopted in the form of Eurocode-2 function [3]. These

The calculation of the strength of the cross sections of bending reinforced concrete elements

≤ f <sup>z</sup><sup>М</sup> f Wz<sup>М</sup>

where σs,Table is the tension in the reinforcement, in which there is no need to determine the

and crack resistance is recommended to be performed according to the formula:

МEd Wc

Calculation of tension limitation in the reinforcement is carried out as follows:

The calculation of the width of the cracks opening is carried out in same scheme:

<sup>σ</sup><sup>z</sup><sup>М</sup> <sup>¼</sup> Ме Wc

width of the cracks opening, which are determined by Table 2.5 [7].

Deflections are determined by curvature by using numerical methods.

mined by the following formulas:

26 Cement Based Materials

tables are given in [6].

• For cross sections without cracks at М < МW:

• For cross sections without cracks at М ≥ М<sup>W</sup>

• For cross sections with a crack in the stretched zone:

The total deformation of fibrous concrete fibers and stretched reinforcement must be deter-

$$
\sigma\_{zM} = M\_\varepsilon / W\_\varepsilon \to \Sigma \varepsilon \to \mathbf{1}/r = \Sigma \varepsilon / d \to f. \tag{33}
$$

A separate important issue in the theory of reinforced concrete is the consideration of regime loads and influences: long-term, quasi-constant, low cycle, temperature, humidity and others. Thus, taking into account the long-term load can be realized by introducing the creep coefficient to the curvature or by introducing into the calculation of the deformation diagrams with the corresponding parameters. The calculation of regime load under the first condition can be carried out according to the given method by using tables for short-term load. When performing calculations under the second condition, it is necessary to use the tables obtained for the corresponding parameters of the diagrams. Similar tables can be made for virtually all regime loads and influences, which greatly simplify the calculations of strength, crack resistance, stiffness and width of crack opening. This is an issue that needs to be studied in detail, but the use of the calculated resistance of reinforced concrete gives confidence in the successful solution of this problem.

## 4. Examples of calculation of bending reinforced concrete elements

Example 1. Reinforced concrete beam with working cross section b � d = 20 � 45 sm is made of concrete of class С25/30 and reinforced 4∅25 of steel of class А500С. Determine the carrying capacity of the beam.

Solution. The percentage of beam reinforcement with stretched reinforcement is calculated:

$$\rho\_f = \frac{A\_s}{bd} \times 100\% = \frac{19.63}{20 \times 45} = 2.181\%.$$

According to the tables the calculated resistance of the reinforced concrete to the bend is determined:

$$f\_{z\text{M}} = 37.12 \text{ MPa}.$$

The carrying capacity of the beam is calculated by the formula:

$$M\_{Ed} = W\_c f\_{zM} = \frac{bd^2}{6} f\_{zM} = \frac{20 \times 45^2}{6} 37.12 \times 10^{-3} = 250.56 \text{ kNm.s}$$

Example 2. Reinforced concrete beam with working cross section b � d = 30 � 45 sm is made of concrete of class С25/30 and steel of class А400С and should take an external moment МEd = 266.46 kNm. Determine element reinforcement.

Solution. The moment of resistance of the concrete cross section is determined:

$$W\_c = \frac{bd^2}{6} = \frac{30 \times 45^2}{6} = 10125 \text{ sm}^3.5$$

<sup>f</sup> <sup>z</sup><sup>М</sup> <sup>¼</sup> MEd Wc

ment will be As <sup>=</sup> <sup>r</sup><sup>f</sup> � <sup>b</sup> � <sup>d</sup> = 0.0142 � <sup>30</sup> � 55 = 23.43 sm2

with known cross-sectional dimensions of concrete.

percentage of reinforcement and the given load.

6. Calculation of the moment of formation of cracks.

7. Calculation the width of the crack opening under operating load.

8. Determination of the deflections of the elements under the operational load.

sectional dimensions of the concrete.

concrete element.

of their calculation.

sm2 is accepted.

5. Conclusions

problems, namely:

<sup>¼</sup> <sup>6</sup>MEd

bd<sup>2</sup> <sup>¼</sup> <sup>6</sup> � <sup>486</sup> � <sup>103</sup>

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their…

From the tables it is clear that such a calculated resistance can be provided for different classes of concrete, reinforcement, and the percentage of reinforcement, starting with the concrete of class С20/25 and percentage of reinforcement 1.5 and more. Following physical, economic, and technological considerations, the designer takes the option that best suits the customer, for example, concrete class С25/30 and reinforcement of class А500С with percentage of reinforcement r<sup>f</sup> = 1.42%. Then the area of the cross section of the working reinforce-

Obtained parameters of the stress-strain state of bending reinforced concrete elements: the calculated resistance of reinforced concrete to bend and tension in the cross section of the bending reinforced concrete element, the above dependencies allow to solve a number of

1. Calculation of the strength of the cross section of the bending reinforced concrete element with known cross-sectional dimensions of the concrete and reinforcement area.

2. Determination of the required cross-sectional area of the reinforcement for a given load

3. Foundation the dimensions of the cross section of concrete and reinforcement for a certain

4. Checking strength with known cross-sectional area of the reinforcement and given cross-

5. Verification of the conditions for ensuring the strength of the cross section of reinforced

Using the calculated resistances of reinforced concrete allowed to reduce the calculation of reinforced concrete elements according to the nonlinear deformation model to the application of the formulas of the classical resistance of materials and to significantly simplify the process

<sup>30</sup> � <sup>552</sup> <sup>¼</sup> <sup>32</sup>:12 MPa:

. By gage 3∅28 + 2∅20, А<sup>s</sup> = 24.75

http://dx.doi.org/10.5772/intechopen.75122

29

The required calculated resistance of the reinforced concrete to the bending is calculated:

$$f\_{zM} = \frac{M\_{Ed}}{W\_c} = \frac{266.46 \times 10^3}{10125} = 26.32 \text{ MPa.}$$

According to the tables the required percentage of reinforcement is determined:

$$\rho\_f = 1.453\%.$$

The area of the cross section of the working reinforcement is equal:

$$A\_s = \rho\_f \times b \times d = 0.01453 \times 30 \times 45 = 19.62 \text{ sm}^2.$$

By gage 4∅25, А<sup>s</sup> = 19.63 sm2 is accepted.

Example 3. Determine the cross-sectional dimensions of the beam of concrete class С16/20 and the area of the cross section of the working reinforcement of steel of class А400С, if the beam perceives the bending moment МEd = 136 kNm, and the contents of the working armature are r<sup>f</sup> = 1.25%.

Solution. According to the tables, the calculated resistance of the reinforced concrete to the bend is found: f<sup>М</sup> = 21.60 MPa. The moment of resistance is determined:

$$\mathcal{W}\_c = \frac{bd^2}{6} = \frac{M\_{Ed}}{f\_{zM}} = \frac{136 \times 10^3}{21.60} = 6296 \text{ sm}^3.$$

Accepting the ratio b = 0.5d, calculate

$$d = \sqrt[3]{12 \times W\_c} = \sqrt[3]{12 \times 6296} = 42.27 \text{ sm.}$$

Accepting b � d = 20 � 42 sm, then the area of cross section of the working reinforcement will be As <sup>=</sup> <sup>r</sup><sup>f</sup> � <sup>b</sup> � <sup>d</sup> = 0.0125 � <sup>20</sup> � 42 = 10.5 sm2 . By gage 2∅20 + 2∅18, А<sup>s</sup> = 11.37 sm<sup>2</sup> .

Example 4. Reinforced concrete beam with working cross section b � d = 30 � 55 sm has to perceive the bending moment МEd = 486 kNm. To define the conditions under which the bearing capacity of the beam will be provided and accept the reinforcement.

Solution. The required calculated resistance of the reinforced concrete to the bend is calculated:

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their… http://dx.doi.org/10.5772/intechopen.75122 29

$$f\_{zM} = \frac{M\_{Ed}}{W\_c} = \frac{6M\_{Ed}}{bd^2} = \frac{6 \times 486 \times 10^3}{30 \times 55^2} = 32.12 \text{ MPa.}$$

From the tables it is clear that such a calculated resistance can be provided for different classes of concrete, reinforcement, and the percentage of reinforcement, starting with the concrete of class С20/25 and percentage of reinforcement 1.5 and more. Following physical, economic, and technological considerations, the designer takes the option that best suits the customer, for example, concrete class С25/30 and reinforcement of class А500С with percentage of reinforcement r<sup>f</sup> = 1.42%. Then the area of the cross section of the working reinforcement will be As <sup>=</sup> <sup>r</sup><sup>f</sup> � <sup>b</sup> � <sup>d</sup> = 0.0142 � <sup>30</sup> � 55 = 23.43 sm2 . By gage 3∅28 + 2∅20, А<sup>s</sup> = 24.75 sm2 is accepted.

### 5. Conclusions

Wc <sup>¼</sup> bd<sup>2</sup>

<sup>f</sup> <sup>z</sup><sup>М</sup> <sup>¼</sup> MEd Wc

The area of the cross section of the working reinforcement is equal:

bend is found: f<sup>М</sup> = 21.60 MPa. The moment of resistance is determined:

<sup>6</sup> <sup>¼</sup> MEd f <sup>z</sup><sup>М</sup>

bearing capacity of the beam will be provided and accept the reinforcement.

<sup>p</sup><sup>3</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Wc <sup>¼</sup> bd<sup>2</sup>

<sup>d</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 � Wc

By gage 4∅25, А<sup>s</sup> = 19.63 sm2 is accepted.

Accepting the ratio b = 0.5d, calculate

be As <sup>=</sup> <sup>r</sup><sup>f</sup> � <sup>b</sup> � <sup>d</sup> = 0.0125 � <sup>20</sup> � 42 = 10.5 sm2

r<sup>f</sup> = 1.25%.

28 Cement Based Materials

lated:

<sup>6</sup> <sup>¼</sup> <sup>30</sup> � <sup>45</sup><sup>2</sup>

The required calculated resistance of the reinforced concrete to the bending is calculated:

<sup>¼</sup> <sup>266</sup>:<sup>46</sup> � <sup>103</sup>

r<sup>f</sup> ¼ 1:453%:

As <sup>¼</sup> <sup>r</sup><sup>f</sup> � <sup>b</sup> � <sup>d</sup> <sup>¼</sup> <sup>0</sup>:<sup>01453</sup> � <sup>30</sup> � <sup>45</sup> <sup>¼</sup> <sup>19</sup>:62 sm2

Example 3. Determine the cross-sectional dimensions of the beam of concrete class С16/20 and the area of the cross section of the working reinforcement of steel of class А400С, if the beam perceives the bending moment МEd = 136 kNm, and the contents of the working armature are

Solution. According to the tables, the calculated resistance of the reinforced concrete to the

<sup>¼</sup> <sup>136</sup> � <sup>103</sup>

<sup>12</sup> � <sup>6296</sup> <sup>p</sup><sup>3</sup>

Accepting b � d = 20 � 42 sm, then the area of cross section of the working reinforcement will

Example 4. Reinforced concrete beam with working cross section b � d = 30 � 55 sm has to perceive the bending moment МEd = 486 kNm. To define the conditions under which the

Solution. The required calculated resistance of the reinforced concrete to the bend is calcu-

<sup>21</sup>:<sup>60</sup> <sup>¼</sup> 6296 sm<sup>3</sup>

¼ 42:27 sm:

:

. By gage 2∅20 + 2∅18, А<sup>s</sup> = 11.37 sm<sup>2</sup>

.

According to the tables the required percentage of reinforcement is determined:

<sup>6</sup> <sup>¼</sup> 10125 sm3

<sup>10125</sup> <sup>¼</sup> <sup>26</sup>:32 MPa:

:

:

Obtained parameters of the stress-strain state of bending reinforced concrete elements: the calculated resistance of reinforced concrete to bend and tension in the cross section of the bending reinforced concrete element, the above dependencies allow to solve a number of problems, namely:


Using the calculated resistances of reinforced concrete allowed to reduce the calculation of reinforced concrete elements according to the nonlinear deformation model to the application of the formulas of the classical resistance of materials and to significantly simplify the process of their calculation.
