Materials

MaterialsConcreteConstitutive lawAllowable stresses and safety factorsFrench national annexBelgian national annexSteelConstitutiveSafety factorsAllowable stressesNotesStressesStrainsReferences

Concrete

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

Allowable stresses and safety factors

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

$f_{ck}^\ast$ 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

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

Steel

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 ‰

$\varepsilon_e$ ) 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‰

Safety factors

ULS
- $\gamma_s=1.15$
- $\gamma_s=1.20$
SLS
- $\gamma_s=1.00$
- $\gamma_s=1.00$

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:
$σ_{s,adm}=1000w_k$
$w_k$ is the limit value on the opening of the cracks (mm)

Notes

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

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

References

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