Materials

1 Concrete

1.1 Constitutive law

The constitutive law of concrete features an elastoplastic behaviour (parabolic-rectangular). $$\begin{array}{cc}\sigma_c=f_{cd}\left[1-\left(1-\frac{\displaystyle\varepsilon_c}{\displaystyle\varepsilon_{c2}}\right)^n\right]&\;0\leq\varepsilon_c\leq\varepsilon_{c2}\\\sigma_c=f_{cd}&\varepsilon_{c2}\leq\varepsilon_c\leq\varepsilon_{cu2}\end{array}$$

Constitutive law of concrete (parabolic-rectangular diagram)

Valeurs numériques des caractéristiques mécaniques des bétons

1.2 Allowable stresses and safety factors

1.2.1 French national annex

In this chapter, we provide allowable stresses and safety factors calculation example for a concrete C30/37, according to NF P94-282 §6.4.

Permissible compressive stress of concrete (characteristic value at 28 days): $$f_{ck}=30MPa$$ Maximal value of characteristic compressive strength during processing (Table 6.4.1.1): $$C_{max}=35MPa$$ Empirical coefficients (Table 6.4.1.1): $$k_1=1.30$$

$$k_2=1.05$$

We consider that integrity checks are not carried out (Table 6.4.1.1), so: $$k_3=1.00$$ Characteristic compressive strength: $$f_{ck}^\ast=\frac{min\left(f_{ck},C_{max}\right)}{k_1k_2}=21.98MPa$$

According to NF P94-282 §6.4.2 (7) Note 1, $f_{ck}^\ast$ should only be used for compressive stress check.

Average compressive strength: $$f_{cm}=f_{ck}+8MPa\;=38MPa$$ Average tensile strength: $$f_{ctm}=0.3f_{ck}^{2/3}\;=2.90MPa$$ Characteristic tensile strength (5%): $$f_{ct;k;0.05}=0.7f_{ctm}\;=2.03MPa$$ Allowable SLS stresses (NF P94-282 §6.4.1 (8)):

• Mean stress:

$$\sigma_{c,mean}=0.3k_3f_{ck}^\ast=6.6MPa$$

• Maximal stress:

$$\sigma_{c,max}=0.6\cdot min\left(k_3f_{ck}^\ast;f_{ck}\right)=13.2MPa$$

Allowable ULS resistances (Eurocode 2 §3.1.6) :

$\alpha_{cc}=1.0$, $\alpha_{ct}=1.0$, $\gamma_c=1.5$

• Compressive:

$$f_{cd}=\alpha_{cc}\frac{min\left(k_3f_{ck}^\ast;C_{max}\right)}{\gamma_c}=14.7MPa$$

• Tensile:

$$f_{ctd}=\alpha_{ct}\frac{f_{ctk,0.05}}{\gamma_c}=1.35MPa$$

Allowable Accidental ULS resistances (Eurocode 2 §3.1.6) :

$\alpha_{cc}=1.0$, $\alpha_{ct}=1.0$, $\gamma_c=1.2$

• Compressive:

$$f_{cd}=\alpha_{cc}\frac{min\left(k_3f_{ck}^\ast;C_{max}\right)}{\gamma_c}=18.3MPa$$

• Tensile:

$$f_{ctd}=\alpha_{ct}\frac{f_{ctk,0.05}}{\gamma_c}=1.7MPa$$

1.2.1.1 Belgian national annex

In this chapter, we provide allowable stresses and safety factors calculation example for a concrete C30/37, according to Belgian national annex (BNA).

Permissible compressive stress of concrete (characteristic value at 28 days): $$f_{ck}=30MPa$$ Maximal value of characteristic compressive strength during processing (Table 6.4.1.1): $$C_{max}=35MPa$$ Empirical coefficient (BNA §2.4.2.5 (2)): $$k_f=1.1$$ We consider that integrity checks are not carried out, so: $$k_3=1.0$$ Characteristic compressive strength: $$f_{ck}^\ast=\frac{min\left(f_{ck},C_{max}\right)}{k_f}=27.27MPa$$ Average compressive strength: $$f_{cm}=f_{ck}+8MPa\;=38MPa$$ Average tensile strength: $$f_{ctm}=0.3f_{ck}^{2/3}\;=2.90MPa$$ Characteristic tensile strength (5%): $$f_{ct;k;0.05}=0.7f_{ctm}\;=2.03MPa$$ Allowable SLS stresses:

• Mean stress:

$$\sigma_{c,mean}=0.3k_3f_{ck}^\ast=8.2MPa$$

• Maximal stress:

$$\sigma_{c,max}=0.6\cdot min\left(k_3f_{ck}^\ast;f_{ck}\right)=16.4MPa$$

Allowable ULS resistances (Eurocode 2 §3.1.6) :

$\alpha_{cc}=0.85$, $\alpha_{ct}=1.0$, $\gamma_c=1.5$

• Compressive:

$$f_{cd}=\alpha_{cc}\frac{min\left(k_3f_{ck}^\ast;C_{max}\right)}{\gamma_c}=15.5MPa$$

• Tensile:

$$f_{ctd}=\alpha_{ct}\frac{f_{ctk,0.05}}{\gamma_c}=1.35MPa$$

Allowable Accidental ULS resistances (Eurocode 2 §3.1.6) :

$\alpha_{cc}=0.85$, $\alpha_{ct}=1.0$, $\gamma_c=1.2$

• Compressive

$$f_{cd}=\alpha_{cc}\frac{min\left(k_3f_{ck}^\ast;C_{max}\right)}{\gamma_c}=19.3MPa$$

• Tensile

$$f_{ctd}=\alpha_{ct}\frac{f_{ctk,0.05}}{\gamma_c}=1.7MPa$$

2 Steel

2.1 Constitutive

The constitutive law of steel features an elastoplastic behaviour.

Steel constitutive law

According to EN 10080, Eurocode 2 defines three ductility classes :

• Class A: normal ductility (welded mesh made of drawn or cold worked wire)

$$\sigma_s=432.71+952.38\varepsilon_S≯454MPa$$

• Class B: high ductility (hot-rolled HA bars)

$$σ_s=433.20+727.27ε_S≯466MPa$$

• Class C: very high ductility (special purpose steels; seismic constructions)

$$σ_s=432.84+895.52ε_S≯493MPa$$

National annex provide ratio between ultimate and characteristic steel strain:

National annex	$\varepsilon_{ud} / \varepsilon_{uk}$
French	0.9
Belgian	0.8

For instance, according to the French National Annex:

Ductility	$\varepsilon_{ud}$	$\varepsilon_{uk}$
A	22.5 ‰	25 ‰
B	45 ‰	50 ‰
C	67.5 ‰	75 ‰

The ultimate strain of the elastic zone ($\varepsilon_e$) depends in any case on the considered combination. In the case of an ULS combination (persistent or transient): $$f_{yd}=\frac{f_{yk}}{\gamma_s}=\frac{500MPa}{1.15}=435MPa$$

$$E_s=200000MPa$$

$$\varepsilon_e=\frac{f_{yd}}{E_s}=\frac{435MPa}{200000MPa}=2.175‰$$

2.2 Safety factors

• ULS
  • Persistent and transient situation: $\gamma_s=1.15$
  • Accidental situation: $\gamma_s=1.20$
• SLS
  • Characteristic or Frequent situation: $\gamma_s=1.00$
  • Quasi permanent situation: $\gamma_s=1.00$

2.3 Allowable stresses

• ULS

$$σ_{s,adm}=f_{yd}=f_{yk}/γ_s$$

• SLS $$σ_{s,adm}=0.8f_{yk}$$ If the control of the opening of cracks is required, it is common to consider: $$σ_{s,adm}=1000w_k$$ where $w_k$ is the limit value on the opening of the cracks (mm)

3 Notes

3.1 Stresses

Symbol	Unit	Description
$f_{ck}$	MPa	Permissible compressive stress of concrete (characteristic value at 28 days)
$C_{max}$	MPa	Characteristic compressive strength during processing
$k_1$	-	Empirical coefficient, depends on the method of concrete pouring in the ground
$k_2$	-	Empirical coefficient, depends on the concrete casting difficulties related to the geometry of the structure
$k_3$	-	Empirical coefficient, depends on whether integrity checks are carried out
$f_{ck}^*$	MPa	Characteristic compressive strength considered in the calculation of concrete walls
$f_{cm}$	MPa	Average compressive strength
$f_{ctm}$	MPa	Average tensile strength
$f_{ct;k;0.05}$	MPa	Characteristic tensile strength (5%)
$f_{cd}$	MPa	Permissible compressive stress of concrete (design value)
$\alpha_{cc}$	-	Coefficient taking into account the long-term effects on the compressive strength of concrete > French national annex = 1.0 > Belgian national annex = 0.85
$\alpha_{ct}$	-	Coefficient taking into account the long-term effects on the tensile strength of concrete > French national annex = 1.0 > Belgian national annex = 1.0
$f_{yd}$	MPa	Permissible stress of steel (design value)
$f_{yk}$	MPa	Permissible stress of steel (characteristic value)
$\gamma_c$	-	Partial coefficient on concrete strength

3.2 Strains

Symbol	Unit	Description
$\varepsilon_{s1}$	-	Strain of the tensioned steel section
$\varepsilon_{s2}$	-	Strain of the compressed steel section
$\varepsilon_c$	-	Concrete strain
$\varepsilon_e$	-	Maximum elastic strain of steel

4 References

• Eurocode 2
• Techniques de l’ingénieur (C 2 330)