General Concepts on Reinforced Concrete

Abstract

This chapter presents the properties of the constitutive materials with their strength parameters and failure criteria. A special discourse is devoted to the creep of concrete and its structural effects. The behaviour of the composite reinforced concrete sections is finally presented with the related basic assumptions for resistance calculations.

The original version of this chapter was revised: For detailed information please see Erratum. The erratum to this chapter is available at 10.1007/978-3-319-52033-9_11

Appendix: Characteristics of Materials

1.1.1 Table 1.1: Hardening Curves of Concrete

The following table shows the values of the ratios f _cj /f _c between the strength at time t from casting and the strength at 28 days, where values deduced from the following formula:

$$ \frac{{f_{{{\text{c}j}}} }}{{f_{\text{c}} }} = {\text{e}}^{{\beta \left( {1 - 1/\sqrt \tau } \right)}}, $$

and the values of the analogous ratio E _cj /E _c between elastic moduli, values deduced from the following formula:

$$ \frac{{E_{{{\text{c}j}}} }}{{E_{\text{c}} }} = \left[ {{\text{e}}^{{\beta \left( {1 - 1/\sqrt \tau } \right)}} } \right]^{0.3} \quad {\text{with}}\;\tau = t/28, $$

where t is expressed in days (t = 0.58 corresponds to about 14 h of ageing, time of possible demoulding of precast elements).

Age	Strengths				Moduli
	Accelerated curing (indicative values)	Concrete			Accelerated curing (indicative values)	Concrete
		Fast setting	Normal setting	Slow setting		Fast setting	Normal setting	Slow setting
Days	β = 0.08	β = 0.20	β = 0.25	β = 0.38	β = 0.08	β = 0.20	β = 0.25	β = 0.38
0.58	0.62	0.30	0.23	0.10	0.87	0.70	0.64	0.51
1	0.71	0.42	0.34	0.20	0.90	0.77	0.72	0.61
2	0.80	0.58	0.50	0.35	0.94	0.85	0.81	0.73
3	0.85	0.66	0.60	0.46	0.95	0.88	0.86	0.79
4	0.88	0.72	0.66	0.54	0.96	0.91	0.88	0.83
5	0.90	0.76	0.71	0.59	0.97	0.92	0.90	0.86
6	0.91	0.79	0.75	0.64	0.97	0.93	0.92	0.88
7	0.92	0.82	0.78	0.68	0.98	0.94	0.93	0.89
10	0.95	0.87	0.85	0.77	0.98	0.96	0.95	0.93
14	0.97	0.92	0.90	0.85	0.99	0.98	0.97	0.95
21	0.99	0.97	0.96	0.94	1.00	0.99	0.99	0.98
28	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
60	1.03	1.07	1.08	1.13	1.01	1.02	1.02	1.04
90	1.04	1.09	1.12	1.18	1.01	1.03	1.03	1.05
180	1.05	1.13	1.16	1.26	1.01	1.04	1.05	1.07
365	1.06	1.16	1.20	1.32	1.02	1.04	1.06	1.09
∞	1.08	1.22	1.28	1.46	1.02	1.06	1.08	1.12

1.1.2 Table 1.2: Strength Classes of Concrete

The following tables show the strength and deformation parameters for different codified classes of concrete, of ordinary and controlled classes. Classes are characterized by characteristic values of cylinder and cubic strengths. Cylinder strength f _c, cubic strength R _c, tensile strength f _ct and elastic modulus E _c are reported in the consecutive columns, indicating the mean values with subscript m and the characteristic values with subscript k. Data are expressed in MPa and are calculated with the following formulas:

$$ \begin{array}{*{20}l} {R_{\text{cm}} = f_{\text{cm}} /0.83} \hfill & {} \hfill \\ {f_{\text{ctm}} = 0.27\sqrt[3]{{f_{\text{m}}^{2} }}} \hfill & {{\text{for}}\;f_{\text{cm}} \le 58} \hfill \\ {f_{\text{ctm}} = 2.12\,\ln \left[ {1 + \left( {f_{\text{cm}} /10} \right)} \right]} \hfill & {{\text{for}}\;f_{\text{cm}} > 58} \hfill \\ {E_{\text{cm}} = 22{,}000\left[ {f_{\text{cm}} /10} \right]^{0.3} } \hfill & {\left( {E_{\text{cm}}^{*} = E_{\text{cm}} /1000} \right)}. \hfill \\ \end{array} $$

In design previsions it is assumed $f_{\text{cm}} = f_{\text{k}} + \Delta f$, with $\Delta f = 8\;{\text{MPa}}$ for ordinary production (common construction sites) and with $\Delta f = 5\;{\text{MPa}}$ for controlled production (prefabrication plants). For the two types of production, it is assumed respectively $f_{\text{ctk}} = 0.7f_{\text{ctm}}$ and ${\text{f}}_{\text{ctk}} = 0.8{\text{f}}_{\text{ctm}}$.

Table 1.2a

Class	Ordinary production—$\Delta f = 8\;{\text{MPa}}$
	Class	$f_{\text{ck}}$	$f_{\text{cm}}$	$R_{\text{cm}}$	$f_{\text{ctm}}$	$f_{\text{ctk}}$	$E_{\text{cm}}^{*}$
Low	C16/20	16	24	29	2.2	1.6	29
	C20/25	20	28	34	2.5	1.7	30
	C25/30	25	33	40	2.8	1.9	31
Medium	C30/37	30	38	46	3.1	2.1	33
	C35/43	35	43	52	3.3	2.3	34
	C40/50	40	48	58	3.6	2.5	35
	C45/55	45	53	64	3.8	2.7	36

Table 1.2b

Class	Controlled production—$\Delta f = 5\;{\text{MPa}}$
	Class	$f_{\text{ck}}$	$f_{\text{cm}}$	$R_{\text{cm}}$	$f_{\text{ctm}}$	$f_{\text{ctk}}$	$E_{\text{cm}}^{*}$
Medium	C30/37	30	35	42	2.9	2.3	32
	C35/43	35	40	48	3.2	2.5	33
	C40/50	40	45	54	3.4	2.7	35
	C45/55	45	50	60	3.7	2.9	36
High	C50/60	50	55	66	3.9	3.1	37
	C55/67	55	60	72	4.1	3.3	38
	C60/75	60	65	78	4.3	3.4	39
	C70/85	70	75	90	4.5	3.6	40

1.1.3 Table 1.3: Deformation Parameters of Concretes

The following table shows the values of the main mechanical characteristics of concrete calculated as a function of the compressive strength with the formulas specified below:

$$ \begin{array}{*{20}l} {E_{\text{c}} = 22000\left[ {f_{\text{c}} /10} \right]^{0.3} } \hfill & {\left( {E_{\text{c}}^{*} = E_{\text{c}} /1000} \right)} \hfill \\ {\kappa = 1.05\,E_{\text{c}} \varepsilon_{\text{c1}} /f_{\text{c}} } \hfill & {} \hfill \\ {\varepsilon_{\text{c1}} = 0.7f_{\text{c}}^{0.31} \times 10^{ - 3} \le 2.8 \times 10^{ - 3} } \hfill & {\left( {\varepsilon_{{{\text{c}}1}}^{*} = 1000\varepsilon_{{{\text{c}}1}} } \right)} \hfill \\ {\varepsilon_{\text{cu}} = \left\{ {2.8 + 27\left[ {\left( {98 - f_{\text{c}} } \right)/100} \right]^{4} } \right\}10^{ - 3} \le 3.5 \times 10^{ - 3} } \hfill & {\left( {\varepsilon_{\text{cu}}^{*} = 1000\varepsilon_{\text{cu}} } \right)} \hfill \\ {f_{\text{ct}} = 0.27\sqrt[3]{{f_{\text{c}}^{2} }}} \hfill & {{\text{for}}\;f_{\text{c}} \le 58} \hfill \\ {f_{\text{ct}} = 2.12\,\ln \left[ {1 + \left( {f_{\text{c}} /10} \right)} \right]} \hfill & {{\text{for}}\;f_{\text{c}} > 58} \hfill \\ {\alpha_{\text{t}} = f_{\text{ct}} /f_{\text{c}} } \hfill & {} \hfill \\ {\kappa_{\text{t}} = 1.05\,E_{\text{c}} \varepsilon_{{{\text{ct}}1}} /f_{\text{ct}} } \hfill & {\left( {\varepsilon_{{{\text{ct}}1}} = 0.00015} \right).} \hfill \\ \end{array} $$

Such values are to be used in the constitutive models σ–ε of concrete in compression and tension, respectively, expressed in the following form:

$$ \begin{array}{*{20}l} {\sigma = \frac{{\kappa \eta - \eta^{2} }}{1 + (\kappa - 2)\eta }f_{\text{c}} } \hfill & {\left( {\eta = \varepsilon /\varepsilon_{{{\text{c}}1}} } \right)} \hfill \\ {\sigma = \left[ {\kappa_{\text{t}} \eta_{\text{t}} - (2\kappa_{\text{t}} - 3)\eta_{\text{t}}^{2} + (\kappa_{\text{t}} - 2)\eta_{\text{t}}^{3} } \right]\alpha_{\text{t}} f_{\text{c}} } \hfill & {\left( {\eta_{\text{t}} = \varepsilon /\varepsilon_{{{\text{ct}}1}} } \right)}. \hfill \\ \end{array} $$

Stresses and elastic moduli are expressed in MPa.

The other deformation characteristics are

• Poisson’s raio v = 0.20

• $\begin{array}{*{20}l} {{\text{coefficient}}\,{\text{of}}\,{\text{thermal}}\,{\text{expansion}}} \hfill & {{\alpha }_{\text{T}} } \hfill = 1.0 \times 10^{ - 5} \,^\circ {\text{C}}^{ - 1}.\\ \end{array}$

$f_{\text{c}}$	$E_{\text{c}}^{*}$	$\kappa$	$\varepsilon_{{{\text{c}}1}}^{*}$	$\varepsilon_{\text{cu}}^{*}$	$f_{\text{ct}}$	${\alpha }_{\text{t}}$	$\kappa_{\text{t}}$
24	28.6	2.35	1.90	3.50	2.25	0.094	2.01
28	30.0	2.21	2.00	3.50	2.49	0.089	1.90
33	31.5	2.07	2.10	3.50	2.78	0.084	1.78
38	32.8	1.96	2.20	3.50	3.05	0.080	1.69
43	34.1	1.87	2.20	3.50	3.31	0.077	1.62
48	35.2	1.79	2.30	3.50	3.57	0.074	1.56
53	36.3	1.72	2.40	3.50	3.81	0.072	1.50
35	32.0	2.03	2.10	3.50	2.89	0.083	1.75
40	33.3	1.92	2.20	3.50	3.16	0.079	1.66
45	34.5	1.84	2.30	3.50	3.42	0.076	1.59
50	35.7	1.76	2.40	3.50	3.66	0.073	1.53
55	36.7	1.70	2.40	3.50	3.90	0.071	1.48
60	37.7	1.64	2.50	3.36	4.13	0.069	1.44
65	38.6	1.59	2.60	3.12	4.27	0.066	1.42
75	40.3	1.50	2.70	2.88	4.54	0.060	1.40

1.1.4 Table 1.4: Drying Shrinkage of Concrete

Drying shrinkage is given by

$$ \varepsilon_{\text{cd}} (t^{\prime } ) = \varepsilon_{{{\text{cd}}\infty }} g_{\text{s}} (t^{\prime } ), $$

where t′ is time expressed in days and measured starting from the onset of the phenomenon.

The following tables show the final value of the drying shrinkage ε _cd∞ for different relative humidities h of the ageing environment, for different strength classes c of concrete and for different equivalent thicknesses s. Values are deduced from the following formula:

$$ \varepsilon_{{{\text{cd}}\infty }} = k_{\text{s}} \varepsilon_{\text{cdo}}, $$

with

$$ \begin{array}{*{20}l} {k_{\text{s}} = 0.7 + 0.0094(5 - s)^{2.5} } \hfill & {{\text{for}}\;s\, < 5} \hfill \\ {k_{\text{s}} = 0.7} \hfill & {{\text{for}}\;s\, < 5} \hfill \\ {\varepsilon_{\text{do}} = 870\left( {1 - h^{3} } \right)e^{{ - 0.12{\text{c}}}} \times 10^{ - 6} }, \hfill & {} \hfill \\ \end{array} $$

where

$h = {\text{RH}}/100$ :

relative humidity ratio;

$c = f_{\text{cm}} /10$ :

mean strength in ${\text{kN}}/{\text{cm}}^{2}$;

$s = \frac{{2A_{\text{c}} /u}}{100}$ :

equivalent thickness in dm;

($A_{\text{c}}$ = cross-sectional area in ${\text{mm}}^{2}$; u = perimeter of the section in mm).

The ones reported in the tables are mean values, for cement of class N and for water/cement ratio ≤0.55, with coefficient of variation of about 0.30. For higher water/cement ratios, shrinkage is greater. For underwater ageing ε _cd∞ = 0.00 can be assumed.

Table 1.4a: Values of $\varepsilon_{{{\text{cd}}\infty }}^{*} = 1000\cdot\varepsilon_{{{\text{cd}}\infty }}$ for RH = 50%

Class	f cm (MPa)	Equivalent thicknesses in mm
		50	100	150	300	≥500
C16/20	24	0.63	0.57	0.52	0.43	0.40
C20/25	28	0.60	0.54	0.50	0.41	0.38
C25/30	33	0.57	0.51	0.47	0.39	0.36
C30/37	38	0.53	0.48	0.44	0.36	0.34
C35/43	43	0.50	0.45	0.42	0.34	0.32
C40/50	48	0.47	0.43	0.39	0.32	0.30
C45/55	53	0.44	0.40	0.37	0.30	0.28
C30/37	35	0.55	0.50	0.46	0.38	0.35
C35/43	40	0.52	0.47	0.43	0.35	0.33
C40/50	45	0.49	0.44	0.41	0.33	0.31
C45/55	50	0.46	0.42	0.38	0.31	0.29
C50/60	55	0.43	0.39	0.36	0.30	0.28
C55/67	60	0.41	0.37	0.34	0.28	0.26
C60/75	65	0.39	0.35	0.32	0.26	0.24
C70/85	75	0.34	0.31	0.28	0.23	0.22

Table 1.4b: Values of $\varepsilon_{{{\text{cd}}\infty }}^{*} = 1000\cdot\varepsilon_{{{\text{cd}}\infty }}$ for RH = 60%

Class	f cm (MPa)	Equivalent thicknesses in mm
		50	100	150	300	≥500
C16/20	24	0.56	0.51	0.47	0.39	0.36
C20/25	28	0.54	0.49	0.45	0.37	0.34
C25/30	33	0.51	0.46	0.42	0.35	0.32
C30/37	38	0.48	0.43	0.40	0.33	0.30
C35/43	43	0.45	0.41	0.37	0.31	0.28
C40/50	48	0.42	0.38	0.35	0.29	0.27
C45/55	53	0.40	0.36	0.33	0.27	0.25
C30/37	35	0.49	0.45	0.41	0.34	0.31
C35/43	40	0.47	0.42	0.39	0.32	0.30
C40/50	45	0.44	0.40	0.36	0.30	0.28
C45/55	50	0.41	0.37	0.34	0.28	0.26
C50/60	55	0.39	0.35	0.32	0.27	0.25
C55/67	60	0.37	0.33	0.30	0.25	0.23
C60/75	65	0.35	0.31	0.29	0.24	0.22
C70/85	75	0.31	0.28	0.25	0.21	0.19

Table 1.4c: Values of $\varepsilon_{{{\text{cd}}\infty }}^{*} = 1000\cdot\varepsilon_{{{\text{cd}}\infty }}$ for RH = 70%

Class	f cm (MPa)	Equivalent thicknesses in mm
		50	100	150	300	≥500
C16/20	24	0.47	0.43	0.39	0.32	0.30
C20/25	28	0.45	0.41	0.37	0.31	0.29
C25/30	33	0.42	0.38	0.35	0.29	0.27
C30/37	38	0.40	0.36	0.33	0.27	0.25
C35/43	43	0.38	0.34	0.31	0.26	0.24
C40/50	48	0.35	0.32	0.29	0.24	0.22
C45/55	53	0.33	0.30	0.28	0.23	0.21
C30/37	35	0.41	0.38	0.34	0.28	0.26
C35/43	40	0.39	0.35	0.32	0.27	0.25
C40/50	45	0.37	0.33	0.30	0.25	0.23
C45/55	50	0.35	0.31	0.29	0.24	0.22
C50/60	55	0.33	0.30	0.27	0.22	0.21
C55/67	60	0.31	0.28	0.25	0.21	0.19
C60/75	65	0.29	0.26	0.24	0.20	0.18
C70/85	75	0.26	0.23	0.21	0.18	0.16

Table 1.4d: Values of $\varepsilon_{{{\text{cd}}\infty }}^{*} = 1000\cdot\varepsilon_{{{\text{cd}}\infty }}$ for RH = 80%

Class	f cm (MPa)	Equivalent thicknesses in mm
		50	100	150	300	≥500
C16/20	24	0.35	0.32	0.29	0.24	0.22
C20/25	28	0.33	0.30	0.28	0.23	0.21
C25/30	33	0.32	0.29	0.26	0.22	0.20
C30/37	38	0.30	0.27	0.25	0.20	0.19
C35/43	43	0.28	0.25	0.23	0.19	0.18
C40/50	48	0.26	0.24	0.22	0.18	0.17
C45/55	53	0.25	0.22	0.21	0.17	0.16
C30/37	35	0.31	0.28	0.26	0.21	0.20
C35/43	40	0.29	0.26	0.24	0.20	0.18
C40/50	45	0.27	0.25	0.23	0.19	0.17
C45/55	50	0.26	0.23	0.21	0.18	0.16
C50/60	55	0.24	0.22	0.20	0.17	0.15
C55/67	60	0.23	0.21	0.19	0.16	0.14
C60/75	65	0.21	0.19	0.18	0.15	0.14
C70/85	75	0.19	0.17	0.16	0.13	0.12

1.1.5 Table 1.5: Drying Shrinkage Curves of Concrete

The following table shows the values of the function g _s(t′) which expresses the time law of drying shrinkage for different values of the equivalent thickness 2A _c/u (A _c = cross-sectional area of concrete; u = its perimeter).

Age	2A c/u (mm)
	Small thickness	Medium–small	Medium thickness	Medium–large	Large thickness
Days	50	100	150	300	600
0.58	0.00	0.00	0.00	0.00	0.00
1	0.23	0.10	0.05	0.02	0.01
2	0.50	0.26	0.16	0.06	0.02
3	0.63	0.38	0.25	0.10	0.04
4	0.71	0.46	0.32	0.14	0.05
5	0.76	0.52	0.38	0.18	0.07
6	0.79	0.58	0.42	0.21	0.08
7	0.82	0.62	0.47	0.24	0.10
10	0.87	0.70	0.56	0.31	0.14
14	0.90	0.77	0.65	0.39	0.19
21	0.94	0.84	0.74	0.50	0.26
28	0.95	0.87	0.79	0.57	0.32
60	0.98	0.94	0.89	0.74	0.50
90	0.98	0.96	0.92	0.81	0.60
180	0.99	0.98	0.96	0.90	0.75
365	1.00	0.99	0.98	0.95	0.86
∞	1.00	1.00	1.00	1.00	1.00

The onset of the phenomenon is assumed at 14 h from casting (t′ = t – 0.58). The values are calculated with the following formula:

$$ g_{\text{s}} = \frac{{t^{\prime } }}{{t^{\prime } + 4\sqrt {s^{3} } }}\quad {\text{with}}\,s = \frac{{2A_{\text{c}} /u}}{100}. $$

For the calculation of shrinkage at time t it can be set as

$$ \varepsilon_{\text{cd}} = \varepsilon_{{{\text{cd}}\infty }} \,g_{\text{s}}, $$

where $\varepsilon_{{{\text{cd}}\infty }}$ is deduced from Table 1.4.

1.1.6 Table 1.6: Autogenous Shrinkage of Concrete

Autogenous shrinkage is given by

$$ \varepsilon_{\text{ca}} (t) = \varepsilon_{{{\text{ca}}\infty }} \,g_{\text{a}} (t), $$

where t is the concrete age expressed in days.

The following table shows the final value of autogenous shrinkage ε _ca∞ for different mean strengths f _cm of concrete. The values are deduced from the following formula:

$$ \varepsilon_{{{\text{ca}}\infty }} = 2.5(f_{\text{cm}} - 18) \cdot 10^{ - 6} $$

(in table $\varepsilon_{{{\text{ca}}\infty }}^{*} = 1000\varepsilon_{{{\text{ca}}\infty }}$).

Ordinary	f cm (MPa)	$\varepsilon_{{{\text{ca}}\infty }}^{*}$
Class
C16/20	24	0.02
C20/25	28	0.03
C25/30	33	0.04
C30/37	38	0.05
C35/43	43	0.06
C40/50	48	0.08
C45/55	53	0.09

Controlled	f cm (MPa)	$\varepsilon_{{{\text{ca}}\infty }}^{*}$
Class
C30/37	35	0.04
C35/43	40	0.06
C40/50	45	0.07
C45/55	50	0.08
C50/60	55	0.09
C55/67	60	0.11
C60/75	65	0.12
C70/85	75	0.14

1.1.7 Table 1.7: Autogenous Shrinkage Curves of Concrete

The following table shows the value of the function g _a(t) that expresses the time law of autogenous shrinkage. The values are calculated with the following formula:

$$ g_{\text{a}} = 1 - e^{ - 0.2\sqrt t }, $$

where t is the concrete age expressed in days starting from casting.

Age	g a
0.58	0.14
1	0.18
2	0.25
3	0.29
4	0.33
5	0.36
6	0.39
7	0.41
10	0.47
14	0.53
21	0.60
28	0.65
60	0.79
90	0.85
180	0.93
365	0.98
∞	1.00

1.1.8 Chart 1.8: Concrete Shrinkage and Nominal Values

Concrete shrinkage is given by

$$ \varepsilon_{{{\text{cs}}\infty }} = \varepsilon_{\text{cd}} + \varepsilon_{\text{ca}}, $$

where

$\varepsilon_{\text{cd}}$ :

is the component of drying shrinkage (Tables 1.4 and 1.5)

$\varepsilon_{\text{ca}}$ :

is the component of autogenous shrinkage (Tables 1.6 and 1.7).

The nominal values of final shrinkage for RH = 60% are reported below for the design of structures, in ordinary and pre-stressed reinforced concrete, as a function of thicknesses, concrete classes and effects to be evaluated.

Type/thickness	Class	Effect	1000 ε cs∞
Ordinary RC structures medium–big	Low	Global deformation	0.38
Ordinary RC structures medium–big	Medium	Global deformation	0.36
Pre-tensioned $\left( {t_{\text{o}} \ge 14\;{\text{ore}}} \right)$ small	High	Pre-stress losses	0.36
Pre-tensioned $\left( {t_{\text{o}} \ge 14\,{\text{ore}}} \right)$ medium–small	High	Pre-stress losses	0.32
Post-tensioned $\left( {t_{\text{o}} \ge 14\,{\text{Gg}}} \right)$ medium	High	Pre-stress losses	0.28

The time of application of pre-stressing is indicated t _o.

1.1.9 Table 1.9: Classes of Consistency of Fresh Concrete

Concerning workability and with reference to the subsidence a of Abrams cone (Slump test), the following classes of consistency of fresh concrete are distinguished.

Denomination	Humid	Plastic	Semi-fluid	Fluid
a (mm)	<50	50 ÷ 100	100 ÷ 150	>150
Class ISO 4103	S1	S2	S3	S4

1.1.10 Chart 1.10: Weight of Concrete Elements

With reference to concrete with normal aggregate, the specific weight of structural elements can be assumed equal to the following nominal values:

• plain concrete 24.0 kN/m³

• reinforced concrete 25.0 kN/m³

(coefficient of variation ≈ 0.06).

1.1.11 Table 1.11: Concrete Production Control

The control charts and the relative diagrams of a continuing concrete production in a given plant are reported below. The charts are to be used following the indications listed below:

• each chart should refer to a homogeneous type of mix constant in time;

• the mix should be named with the class and with a market specification of the final product;

• basic data should be added (content of cement, water/cement ratio, admixture content and aggregate size);

• the type of curing should be specified, also via the evaluation of β of the hardening law (see Table 1.1);

• it has to be specified whether strength measurements are referred to the reference age (28 days) f or at earlier ages f _j ;

• 28-day tests should always be carried, tests at earlier ages only if required by early stages verifications;

• the chart is made of consecutive sheets, one for each solar month, where normally each row corresponds to a day;

• one concrete sample has to be taken every production day and cured in the same environment of casting;

• a sample consists of two specimens for 28-day tests, plus two specimens for earlier ages’ tests if required;

• data, written on the row of the day of sampling, should start with the date of test;

• the strength measurements of the two specimens and the mean value should then be reported;

• if measured on cubic specimens, the strength value should be reduced with a factor of 0.83 to obtain the cylinder strength f _j ;

• the mean value f _j should be corrected based on the age j of the specimen to deduce the reference (28 days) strength;

• the statistics should be calculated with the values of the set of n samples available in the last 21 solar days;

• for sets of n < 6 samples a conventional deviation of ks = 8 MPa should be assumed;

• for sets of 6 ≤ n ≤ 15 samples the value of k should be taken from the table reported further on;

• for sets of 16 ≤ n ≤ 21 samples the fixed value of k = 1.48 is assumed;

• for the n measurements available, the mean value f _m and the standard deviation s are then calculated;

• the current characteristic strength f _k is finally deduced, to be compared with one of the expected classes.

The formulas for the required calculations are (where R ₁ and R ₂ are the cubic strengths of the two cubic specimens and t is the concrete age in days at the time of testing):

$$ \begin{aligned} f_{j} = 0.83\frac{{R_{1} + R_{2} }}{2}\quad f = \frac{{f_{j} }}{{{\text{e}}^{\beta (1 - 1/\tau )} }}\quad \tau = t/28 \hfill \\ f_{\text{m}} = \frac{{\sum\nolimits_{i = 1}^{n} {f_{i} } }}{n}\quad s = \frac{{\sqrt {\sum\nolimits_{i = 1}^{n} {\left( {f_{i} - f_{\text{m}} } \right)^{2} } } }}{n - 1}\quad f_{\text{k}} = f_{\text{m}} - ks. \hfill \\ \end{aligned} $$

n	6	7	8	9	10	11	12	13	14	15
k	1.87	1.77	1.72	1.67	1.62	1.58	1.55	1.52	1.50	1.48

In any case the values of the variation coefficients s/f _m shall be less than 0.15.

The following pages contain

• the template of control chart for data recording (with values shown as example);

• the diagram relative to the results of testing for the visualization of the production trend (with marks shown as example using, for decimals, the comma instead of the point following the European praxis).

1.1.12 Table 1.12: Creep: Class Coefficient

The following table shows, for the different strength classes of concrete, the value of the coefficient β _c of the formula:

$$ {\varphi }_{\infty } = {\beta }_{\text{c}} {\beta }_{\text{hs}} \,{\varphi }_{\text{o}} $$

for the calculation of final concrete creep. The values are calculated with

$$ \beta_{\text{c}} = \frac{1.673}{\sqrt c }, $$

where c = f _c/10 is the class index and f _c is the mean strength in MPa.

For the other coefficients of the formulas, one can refer to Tables 1.13 and 1.14.

Ordinary	f c (MPa)	β c
Class
C16/20	24	1.08
C20/25	28	1.00
C25/30	33	0.92
C30/37	38	0.86
C35/43	43	0.81
C40/50	48	0.76
C45/55	53	0.73

Controlled	f c (MPa)	β c
Class
C30/37	35	0.89
C35/43	40	0.84
C40/50	45	0.79
C45/55	50	0.75
C50/60	55	0.71
C55/67	60	0.68
C60/75	65	0.66
C70/85	75	0.61

1.1.13 Table 1.13: Creep: Ambient Coefficient

The following table shows, for the different relative humidities RH of the ageing environment and for the different equivalent thicknesses 2A _c/u, the values of the coefficient β _hs of the following formula:

$$ {\varphi }_{\infty } = {\beta }_{\text{c}} {\beta }_{\text{hs}} \,{\varphi }_{\text{o}} $$

for the calculation of concrete final creep. The values are calculated with

$$ \beta_{\text{hs}} = 0.725\left[ {1 + \frac{1 - h}{{0.46\sqrt[3]{s}}}} \right] $$

where h = HR/100 and s = (2A _c/u)/100 (A _c = cross-sectional area of concrete; u = its perimeter).

For the other coefficients of the formula, one can refer to Tables 1.12, 1.13, 1.14.

Relative humidity	2A c/u (mm)
	Small thickness	Medium–small	Medium thickness	Medium–big	Big thickness
%	50	100	150	300	600
80	1.12	1.04	1.00	0.94	0.90
70	1.32	1.20	1.14	1.05	0.98
60	1.52	1.35	1.28	1.16	1.07
50	1.72	1.51	1.41	1.27	1.16

1.1.14 Table 1.14: Creep: Reference Coefficient

The following table shows, for the different concrete ages at loading, the values of the coefficient φ _o of the following formula:

$$ {\varphi }_{\infty } = {\beta }_{\text{c}} {\beta }_{\text{hs}} \,{\varphi }_{\text{o}}, $$

for the calculation of final creep. The values are calculated with

$$ {\varphi }_{\text{o}} = \frac{4.37}{{0.1 + {t}_{\text{o}}^{0.2} }}\quad ({t}_{\text{o}} \,{\text{in}}\,{\text{days}}), $$

and should be assumed, with a coefficient of variation of about 0.20, for water/cement ratios ${\le}0.55$. For higher ratios, the values are greater.

For the definition of t _o see Table 1.15; for the other coefficients of the formula, see Tables 1.12 and 1.13.

Age	φ o
0.58	4.38
1	3.97
2	3.50
3	3.25
4	3.08
5	2.95
6	2.85
7	2.77
10	2.59
14	2.43
21	2.25
28	2.13
60	1.85
90	1.71
180	1.49
365	1.30

1.1.15 Table 1.15: Creep: Effect of Temperature

The following table shows, as a function of the average temperature θ of concrete in the time interval $0 - \bar{t}_{\text{o}}$, the value of the correction factor β _T with which the nominal age t _o can be deduced from the effective age $\bar{t}_{\text{o}}$ at loading:

$$ t_{\text{o}} = \beta_{\text{T}} \bar{t}_{\text{o}} $$

This nominal age is used in the formula φ _o = φ _o(t _o) of creep (see Table 1.14). The values are calculated with the following formula:

$$ \beta_{\text{T}} = e^{{\left( {13.65 - \frac{4000}{273 + \theta }} \right)}} \quad (\theta \;{\text{in}}\;\,^\circ {\text{C}}) $$

θ	β T
10	0.62
15	0.79
20	1.00
25	1.26
30	1.57
35	1.34
40	2.39
45	2.92
50	3.55
55	4.28
60	5.14
65	6.15
70	7.30
75	8.63

1.1.16 Table 1.16: Creep: Nominal Coefficients

The nominal final values of creep coefficients are given below, for the design of ordinary and pre-stressed reinforced concrete, calculated in prevision of an environment with HR = 60% of relative humidity.

Type/thickness	Concrete class	Curing/age	Calculated effect	φ ∞
Ordinary RC struct. medium–big	Low	Natural $\bar{t}_{\text{o}} \ge 14$ days	Global deformation	3.1
Ordinary RC struct. medium–big	Medium	Natural $\bar{t}_{\text{o}} \ge 14$ days	Global deformation	2.5
Pre-tensioned small	High	Accelerated $\bar{t}_{\text{o}} \ge 14$ h	Pre-stress losses	3.1
Pre-tensioned medium–small	Medium	Accelerated $\bar{t}_{\text{o}} \ge 14$ h	Pre-stress losses	2.7
Post-tensioned medium	Medium	Natural $\bar{t}_{\text{o}} \ge 14$ days	Pre-stress losses	1.9

1.1.17 Table 1.17: Characteristics of Reinforcing Steel

B450C steel, used in reinforced concrete structures, is characterized by the following nominal values of characteristic yield strength f _yo and ultimate strength f _to

$$ \begin{aligned} f_{\text{yo}} = 450\;{\text{MPa}} \hfill \\ f_{\text{to}} = 540\;{\text{MPa}} \hfill \\ \end{aligned}$$

The following table shows the requirements for the actual values of the main mechanical characteristics of B450C steel:

Characteristics	Symbol	Value
Characteristic yield strength (fractile 5%)	f yk	${\ge}450\,{\text{MPa}}$
Characteristic ultimate strength (fractile 5%)	f tk	${\ge}540\,{\text{MPa}}$
Uniform elongation (fractile 10%) (=εuk)	(A gt)k	${\ge}7.5\%$
Strain-hardening ratio	(f t/f y)k
Minimum (fractile 10%)		${\ge}1.15$
Maximum (fractile 10%)		${\le}1.35$
Overstrength ratio (fractile 10%)	(f t/f yo)k	${\le}1.25$

Bars and wires made of B450C steel have to be bendable and weldable. Other characteristics common for all types of steel are

• $\begin{array}{*{20}l} {{\text{specific}}\,{\text{weight}}\,\left( {\text{density}} \right)} \hfill & {{\text{g}} = 7 8 50\;{\text{kg}}/{\text{m}}^{ 3} } \hfill \\ \end{array}$

• $\begin{array}{*{20}l} {{\text{longitudinal}}\,{\text{elastic}}\,{\text{modulus}}} \hfill & {{\text{E}}_{\text{s}} = 205000\;{\text{MPa}}} \hfill \\ \end{array}$

• $\begin{array}{*{20}l} {{\text{coefficient}}\,{\text{of}}\,{\text{thermal}}\,{\text{expansion}}} \hfill & {{\alpha }_{\text{T}} = 1.0 \times 10^{ - 5} \;^\circ {\text{C}}^{ - 1} }. \hfill \\ \end{array}$

1.1.18 Table 1.18: Bars and Wires: Commercial Diameters

ϕ (mm)	g (kg/m)	u (mm)	nA s (mm2)
			1	2	3	4	5	6	7	8	9
6	0.222	18.9	28.3	56.5	84.8	113	141	170	198	226	254
8	0.395	25.1	50.5	101	151	201	251	302	352	402	452
10	0.617	31.4	79.0	157	236	314	393	471	550	628	707
12	0.888	37.7	113	226	339	452	566	679	791	905	1131
14	1.208	44.0	154	308	462	616	770	924	1078	1232	1385
16	1.578	50.3	201	402	603	804	1005	1206	1407	1608	1810
18*	1.998	56.6	254	509	763	1018	1272	1527	1781	2036	2290
20	2.466	62.8	314	628	942	1257	1571	1885	2199	2513	2827
22*	2.984	69.1	380	760	1140	1521	1901	2281	2661	3041	3421
24*	3.551	75.4	452	905	1357	1810	2262	2714	3167	3619	4072
25	3.853	78.5	491	982	1473	1963	2454	2945	3436	3927	4418
26*	4.168	81.7	531	1062	1593	2124	2655	3186	3717	4247	4778
28	4.834	88.0	616	1232	1847	2463	3079	3695	4310	4926	5542
30	5.559	94.3	707	1414	2121	2827	3534	4241	4948	5655	6362
32	6.313	100.5	804	1608	2413	3218	4022	4827	5631	6436	7240

1. Note Non-standard diameters are in italic; the diameters not normalized at European level (EN10080) are marked with a star

The table gives the weight g, the perimeter u and the cross-sectional area A _s for the commercial diameters ϕ of the hot-rolled ribbed wires and bars for reinforced concrete. Bars are supplied in 12-m-long bundles, wires up to diameters of 12 mm can be supplied in rolls.

1.1.19 Table 1.19: Bars for PC: Standard Diameters

The following table shows, for nominal diameters ϕ normalized by the European standard EN 10138/4, the values of

g :

unit weight

u :

perimeter of the equivalent bar

A _p :

cross-sectional area

f _ptk :

characteristic rupture strength

f _0.1k :

characteristic strength at 0.1% residual elongation

(f _0.1/f _pt)_k :

hardening (reverse) ratio (=α_rk)

ε _uk :

indicative value of ultimate elongation

F _ptk :

characteristic value of rupture load

F _0.1k :

characteristic value of load at 0.1% residual elongation.

There are two types of steel Fe1030 and Fe1230 produced in hot-rolled bars subsequently subjected to cold-forming.

For the considered types of steel the following standard requirements are applied:

$$ {\varepsilon }_{\text{uk}} \ge 3.5\% \quad {\alpha }_{\text{rk}} \ge 0.80. $$

The other general characteristics of the type of product are

• $\begin{array}{*{20}l} {{\text{specific}}\,{\text{weight}}\,\left( {\text{density}} \right)} \hfill & {g = 7 8 50\,{\text{kg}}/{\text{m}}^{ 3} } \hfill \\ \end{array}$

• $\begin{array}{*{20}l} {{\text{longitudinal}}\,{\text{elastic}}\,{\text{modulus}}} \hfill & {E_{\text{p}} = 20 5000\,{\text{MPa}}} \hfill \\ \end{array}$

• $\begin{array}{*{20}l} {{\text{coefficient}}\,{\text{of}}\,{\text{thermal}}\,{\text{expansion}}} \hfill & {\alpha_{\text{T}} = 1.0 \times 10^{ - 5} \;^\circ {\text{C}}^{ - 1} }. \hfill \\ \end{array}$

ϕ (mm)	g (kg/m)	u (mm)	A p (mm2)	f ptk (MPa)	f 0.1k (MPa)	α rk	ε uk (%)	F ptk (kN)	F 0.1k (kN)
20 20	2.47	62.8	314	1030 1230	830 1080	0.81 0.88	6.0 5.0	325 385	260 340
25 25	3.86	78.5	491	1030 1230	830 1080	0.81 0.88	6.0 5.0	505 600	416 530
26 26	4.17	81.7	531	1030 1230	830 1080	0.81 0.88	6.0 5.0	547 653	443 575
32 32	6.31	101	804	1030 1230	830 1080	0.81 0.88	6.0 5.0	830 870	670 1109
36 36	7.99	113	1018	1030 1230	830 1080	0.81 0.88	6.0 5.0	1050 1100	1208 1400
40 40	9.86	126	1257	1030 1230	830 1080	0.81 0.88	6.0 5.0	1295 1357	1050 1732
50	15.5	157	1960	1030	830	0.81	6.0	2020	1636

For the two types of steel in smooth and ribbed bars, the following table gives the values of

${\delta } = 100\left( {{\text{f}}_{\text{ptm}} - {f}_{\text{ptk}} } \right)/{f}_{\text{ptm}}$ :

percent deviation

${\Updelta \bar{\sigma }}$ :

fatigue limit range for 2 × 10⁶ loading cycles.

Type	δ (%)	${\Updelta \bar{\sigma }}\,{\text{(MPa)}}$
Fe1030	≈7.5	200	Smooth
		180	Ribbed
Fe1230	≈6.0	200	Smooth
		180	Ribbed

1.1.20 Table 1.20: Cold-Drawn Wire: Standard Diameters

The following table shows, for the nominal diameters ϕ normalized by the European standard EN 10138/2, the values of

g :

unit weight

u :

perimeter of the equivalent bar

A _p :

cross-sectional area

f _ptk :

characteristic rupture strength

f _0.1k :

characteristic strength at 0.1% residual elongation

(f _0.1/f _pt)_k :

hardening (reverse) ratio (=α _rk)

ε _uk :

indicative value of ultimate elongation

F _ptk :

characteristic value of rupture load

F _0.1k :

characteristic value of load at 0.1% residual elongation.

There are four types of steels, namely Fe1570, Fe1670, Fe1770 and Fe1870, produced in smooth or indented wires by cold drawing and stretching.

For the considered steels the following standard requirements are applied:

$$ {{\varepsilon }_{\text{uk}} \ge 3.5\% } \quad {{\alpha }_{\text{rk}} \ge 0.80}. $$

The other general characteristics of the type of products are

• $\begin{array}{*{20}l} {{\text{specific}}\,{\text{weight}}\,\left( {\text{density}} \right)} \hfill & {g = 7850\,{\text{kg}}/{\text{m}}^{3} } \hfill \\ \end{array}$

• $\begin{array}{*{20}l} {{\text{longitudinal}}\,{\text{elastic}}\,{\text{modulus}}} \hfill & {E_{\text{p}} = 205000\,{\text{MPa}}} \hfill \\ \end{array}$

• $\begin{array}{*{20}l} {{\text{coefficient}}\,{\text{of}}\,{\text{thermal}}\,{\text{expansion}}} \hfill & {\alpha_{\text{T}} = 1.0 \times 10^{ - 5} \,^{ \circ } {\text{C}}^{ - 1} }. \hfill \\ \end{array}$

The value of deviation $\delta = 100(f_{\text{ptm}} - f_{\text{ptk}} )/f_{\text{ptm}}$ is for all types of steel ${\delta } \cong {7}{.5\% }$.

The fatigue limit range for $2 \cdot 10^{6}$ loading cycles is

$$ \begin{aligned} {\Updelta \bar{\sigma }} & = 200\,{\text{MPa}}\quad {\text{for}}\,{\text{smooth}}\,{\text{wires}} \\ {\Updelta \bar{\sigma }} & = 180\,{\text{MPa}}\quad {\text{for}}\,{\text{indented}}\,{\text{wires}}. \\ \end{aligned} $$

ϕ (mm)	g (kg/m)	u (mm)	A p (mm2)	f ptk (MPa)	f 0.1k (MPa)	α rk	ε uk (%)	F ptk (kN)	F 0.1k (kN)
4.0 4.0	0.989	12.6	12.6	1770 1860	1520 1600	0.86 0.86	4.2 4.0	22.3 23.4	19.2 20.1
5.0 5.0	0.154	15.7	19.6	1670 1770	1440 1520	0.86 0.86	4.6 4.2	32.7 34.7	28.1 29.8
6.0 6.0	0.222	18.9	28.3	1670 1770	1440 1520	0.86 0.86	4.6 4.2	47.3 50.1	40.7 43.1
7.0	0.302	22.0	38.5	1670	1440	0.86	4.6	64.3	55.3
7.5	0.347	23.6	44.2	1670	1440	0.86	4.6	73.8	63.5
8.0	0.395	25.1	50.3	1670	1440	0.86	4.6	84.0	72.2
9.4	0.545	29.5	69.4	1570	1300	0.83	5.0	109.0	90.5
10.0	0.616	31.4	78.5	1570	1300	0.83	5.0	123.0	102

1.1.21 Table 1.21: Strands: Standard Diameters

The following table shows, for the nominal diameters ϕ normalized by the European standard EN 10138/3, the values of

g :

unit weight

u :

perimeter of the equivalent bar

A _p :

cross-sectional area

f _ptk :

characteristic rupture strength

f _0.1k :

characteristic strength at 0.1% residual elongation

(f _0.1/f _pt)_k :

hardening (reverse) ratio (=α _rk)

ε _uk :

indicative value of ultimate elongation

F _ptk :

characteristic value of rupture load

F _0.1k :

characteristic value of load at 0.1% residual elongation.

There are strands made of three wires 3W, seven wires 7W and compacted strands of seven wires 7WC obtained from cold-drawn wires of small diameters (2.4 ÷ 6.0 mm), in six types of steels, namely Fe1700, Fe1770, Fe1820, Fe1860, Fe1960 and Fe2060.

For the concerned steels there are the following standard requirements:

$$ {{\varepsilon }_{\text{uk}} \ge 3.5\% } \quad {{\alpha }_{\text{rk}} \ge 0.80} $$

The other general characteristics of the type of products are

• $\begin{array}{*{20}l} {{\text{specific}}\,{\text{weight}}\,\left( {\text{density}} \right)} \hfill & {g = 7850\,{\text{kg}}/{\text{m}}^{3} } \hfill \\ \end{array}$

• $\begin{array}{*{20}l} {{\text{longitudinal}}\,{\text{elastic}}\,{\text{modulus}}} \hfill & {E_{\text{p}} = 195000\,{\text{MPa}}} \hfill \\ \end{array}$

• $\begin{array}{*{20}l} {{\text{coefficient}}\,{\text{of}}\,{\text{thermal}}\,{\text{expansion}}} \hfill & {\alpha_{\text{T}} = 1.0 \times 10^{ - 5} \,^{ \circ } {\text{C}}^{ - 1} }. \hfill \\ \end{array}$

The value of deviation $\delta = {100}(f_{\text{ptm}} - f_{\text{ptk}} )/f_{\text{ptm}}$ is for all steels 7.5%.

The fatigue limit range for 2 × 10⁶ loading cycles is

$$ \begin{aligned} {\Updelta \bar{\sigma }} & = 190\,{\text{MPa}}\quad {\text{for}}\,{\text{smooth}}\,{\text{wires}} \\ {\Updelta \bar{\sigma }} & = 170\,{\text{MPa}}\quad {\text{for}}\,{\text{indented}}\,{\text{wires}}. \\ \end{aligned} $$

ϕ (mm)	g (kg/m)	u (mm)	A p (mm2)	f ptk (MPa)	f 0.1k (MPa)	α rk	ε uk (%)	F ptk (kN)	F 0.1k (kN)
Strand 3W
5.2	0.107	16.3	13.6	1960	1670	0.85	4.6	26.7	22.7
5.2	0.107	16.3	13.6	2060	1750	0.85	4.2	28.0	23.8
6.5	0.166	20.4	21.2	1860	1580	0.85	4.6	39.4	33.5
6.5	0.166	20.4	21.2	1960	1670	0.85	4.6	41.5	35.3
6.8	0.184	21.4	23.4	1860	1580	0.85	4.6	43.5	37.0
7.5	0.228	23.6	29.0	1860	1580	0.85	4.6	53.9	45.8
Strand 7W
7.0	0.236	22.0	30.0	2060	1750	0.85	4.6	61.8	52.5
9.0	0.393	28.3	50.0	1860	1580	0.85	5.0	93.0	79.0
11.0	0.590	34.6	75.0	1860	1580	0.87	5.0	139	118
12.5	0.730	39.3	93.0	1860	1580	0.85	5.0	173	147
13.0	0.785	40.8	100	1860	1580	0.85	5.0	186	158
15.2	1.090	47.8	139	1770	1500	0.85	5.0	246	209
15.2	1.090	47.8	139	1860	1580	0.85	5.0	258	219
16.0	1.180	50.3	150	1770	1500	0.85	5.0	265	225
16.0	1.180	50.3	150	1860	1580	0.85	5.0	279	237
18.0	1.570	56.5	200	1770	1500	0.85	5.0	354	301
Compacted 7WC
12.7	0.890	40.0	112	1860	1580	0.85	5.0	209	178
15.2	1.295	47.8	165	1820	1580	0.85	5.0	300	225
18.0	1.750	56.5	223	1700	1580	0.85	5.0	380	323

1.1.22 Chart 1.22: Concrete σ–ε Models

For the analysis of a section in reinforced or pre-stressed concrete at the ultimate limit state of rupture, one of the three models σ–ε for concrete described below can be adopted (see also Fig. 1.28).

Classes up to C50/60 ( f _ck ≤ 50 MPa)

For all models,

• ultimate compressive strain of the most stressed fibre $\varepsilon_{\text{cu}} = 0.35\%$

• mean ultimate strain of concrete in compression $\varepsilon_{\text{c2}} = 0.20\%$

• compressive strength of concrete $f_{\text{cd}} = \alpha_{\text{cc}} f_{\text{ck}} /\gamma_{\text{C}}$

• tensile strength of concrete $f_{\text{ctd}} = 0.$

Parabola–rectangle model

$$ \begin{array}{*{20}l} {{\sigma }_{\text{c}} = [1 - (1 - {\varepsilon }_{\text{c}} /{\varepsilon }_{{{\text{c}2}}} )^{2} ]\,{f}_{\text{cd}} } \hfill & {{\text{for}}\,0 \le {\varepsilon }_{\text{c}} < {\varepsilon }_{{{\text{c}2}}} } \hfill \\ {{\sigma }_{\text{c}} = {f}_{\text{cd}} } \hfill & {{\text{for}}\,{\varepsilon }_{{{\text{c}2}}} \le {\varepsilon }_{\text{c}} \le {\varepsilon }_{\text{cu}} } \hfill \\ \end{array} $$

with $\varepsilon_{{{\text{c}}2}} = 0.2\%$.

Triangle–rectangle model

$$ \begin{array}{*{20}l} {{\sigma }_{\text{c}} = ({\varepsilon }_{\text{c}} /{\varepsilon }_{{{\text{c}3}}} )\,{\text{f}}_{\text{cd}} } \hfill & {{\text{for}}\,0 \le {\varepsilon }_{\text{c}} { < \varepsilon }_{{{\text{c}3}}} } \hfill \\ {{\sigma }_{\text{c}} = {\text{f}}_{\text{cd}} } \hfill & {{\text{for}}\,{\varepsilon }_{{{\text{c}3}}} \le {\varepsilon }_{\text{c}} \le {\varepsilon }_{\text{cu}} } \hfill \\ \end{array} $$

with $\varepsilon_{{{\text{c}}3}} = 0.15\%$.

Rectangular model

$$ \begin{array}{*{20}l} {{\sigma }_{\text{c}} = ({\varepsilon }_{\text{c}} /{\varepsilon }_{{{\text{c}3}}} )\,{f}_{\text{cd}} } \hfill & {{\text{for}}\,0 \le {\varepsilon }_{\text{c}} { < \varepsilon }_{{{\text{c}3}}} } \hfill \\ {{\sigma }_{\text{c}} = {\text{f}}_{\text{cd}} } \hfill & {{\text{for}}\,{\varepsilon }_{{{\text{c}3}}} \le {\varepsilon }_{\text{c}} \le {\varepsilon }_{\text{cu}} } \hfill \\ \end{array} $$

with $\varepsilon_{{{\text{c}}4}} = 0.07\% ( = 0.2\varepsilon_{\text{cu}} )$.

Classes greater than C50/60 ( f _ck  > 50 MPa)

For all models,

• ultimate compressive strain of the most stressed fibre

  $\varepsilon_{\text{cu}} = 0.26 + 3.5[(90 - f_{\text{ck}} )/100]^{4} \%$

• mean ultimate strain of concrete in compression

  $\varepsilon_{{{\text{c}}2}} = 0.20 + 0.0085(f_{\text{ck}} - 50)^{0.53} \%$

• compressive strength of concrete $f_{\text{cd}} = \alpha_{\text{cc}} f_{\text{ck}} /\gamma_{\text{C}}$

• tensile strength of concrete $f_{\text{ctd}} = 0.$

Parabola–rectangle model

$$ \begin{array}{*{20}l} {{\sigma }_{\text{c}} = [1 - (1 - {\varepsilon }_{\text{c}} /{\varepsilon }_{{{\text{c}2}}} )^{\text{n}} ]\,{f}_{\text{cd}} } \hfill & {{\text{for}}\,{0} \le {\varepsilon }_{\text{c}} < {\varepsilon }_{\text{cr}} } \hfill \\ {{\sigma }_{\text{c}} = {\text{f}}_{\text{cd}} } \hfill & {{\text{for}}\,{\varepsilon }_{\text{cr}} \le {\varepsilon }_{\text{c}} < {\varepsilon }_{\text{cu}} } \hfill \\ \end{array} $$

with $n = 1.4 + 23.4[(90 - f_{\text{ck}} )/100]^{4}$.

Triangle–rectangle model

$$ \begin{array}{*{20}l} {{\sigma }_{\text{c}} = {\varepsilon }_{\text{c}} /{\varepsilon }_{{{\text{c}}3}} \,{f}_{\text{cd}} } \hfill & {{\text{for}}\,{0} \le {\varepsilon }_{\text{c}} < {\varepsilon }_{{{\text{c}3}}} } \hfill \\ {{\sigma }_{\text{c}} = {f}_{\text{cd}} } \hfill & {{\text{for}}\,{\varepsilon }_{{{\text{c}3}}} \le {\varepsilon }_{\text{c}} \le {\varepsilon }_{\text{cu}} } \hfill \\ \end{array} $$

with $\varepsilon_{{{\text{c}}3}} = 0.15 + 0.55[(f_{\text{ck}} - 50)/400]$.

Rectangular model

$$ \begin{array}{*{20}l} {{\sigma }_{\text{c}} = 0} \hfill & {{\text{for}}\,{0} \le {\varepsilon }_{\text{c}} < {\varepsilon }_{{{\text{c}}4}} } \hfill \\ {{\sigma }_{\text{c}} = {\eta }\,{f}_{\text{cd}} } \hfill & {{\text{for}}\,{\varepsilon }_{{{\text{c}}4}} \le {\varepsilon }_{\text{c}} \le {\varepsilon }_{\text{cu}} } \hfill \\ \end{array} $$

with $\varepsilon_{{{\text{c}}4}} = \lambda \varepsilon_{\text{cu}}$ and

$$ \begin{aligned} \eta = 1.0 - (f_{\text{ck}} - 50)/200 \hfill \\ \lambda = 0.2 + (f_{\text{ck}} - 50)/400 \hfill \\ \end{aligned} $$