Materials

Concrete

Constitutive law

The constitutive law of concrete features an elastoplastic behaviour (parabolic-rectangular).

(1)σc=fcd[1(1εcεc2)n]0εcεc2σc=fcdεc2εcεcu2

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):

(2)fck=30MPa

Maximal value of characteristic compressive strength during processing (Table 6.4.1.1):

(3)Cmax=35MPa

Empirical coefficients (Table 6.4.1.1):

(4)k1=1.30
(5)k2=1.05

We consider that integrity checks are not carried out (Table 6.4.1.1), so:

(6)k3=1.00

Characteristic compressive strength:

(7)fck=min(fck,Cmax)k1k2=21.98MPa

According to NF P94-282 §6.4.2 (7) Note 1, fck should only be used for compressive stress check.

Average compressive strength:

(8)fcm=fck+8MPa=38MPa

Average tensile strength:

(9)fctm=0.3fck2/3=2.90MPa

Characteristic tensile strength (5%):

(10)fct;k;0.05=0.7fctm=2.03MPa

Allowable SLS stresses (NF P94-282 §6.4.1 (8)):

(11)σc,mean=0.3k3fck=6.6MPa
(12)σc,max=0.6min(k3fck;fck)=13.2MPa

 

Allowable ULS resistances (Eurocode 2 §3.1.6) :

αcc=1.0, αct=1.0, γc=1.5

(13)fcd=αccmin(k3fck;Cmax)γc=14.7MPa
(14)fctd=αctfctk,0.05γc=1.35MPa

Allowable Accidental ULS resistances (Eurocode 2 §3.1.6) :

αcc=1.0, αct=1.0, γc=1.2

(15)fcd=αccmin(k3fck;Cmax)γc=18.3MPa
(16)fctd=αctfctk,0.05γc=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):

(17)fck=30MPa

Maximal value of characteristic compressive strength during processing (Table 6.4.1.1):

(18)Cmax=35MPa

Empirical coefficient (BNA §2.4.2.5 (2)):

(19)kf=1.1

We consider that integrity checks are not carried out, so:

(20)k3=1.0

Characteristic compressive strength:

(21)fck=min(fck,Cmax)kf=27.27MPa

Average compressive strength:

(22)fcm=fck+8MPa=38MPa

Average tensile strength:

(23)fctm=0.3fck2/3=2.90MPa

Characteristic tensile strength (5%):

(24)fct;k;0.05=0.7fctm=2.03MPa

Allowable SLS stresses:

(25)σc,mean=0.3k3fck=8.2MPa
(26)σc,max=0.6min(k3fck;fck)=16.4MPa

 

Allowable ULS resistances (Eurocode 2 §3.1.6) :

αcc=0.85, αct=1.0, γc=1.5

(27)fcd=αccmin(k3fck;Cmax)γc=15.5MPa
(28)fctd=αctfctk,0.05γc=1.35MPa

Allowable Accidental ULS resistances (Eurocode 2 §3.1.6) :

αcc=0.85, αct=1.0, γc=1.2

(29)fcd=αccmin(k3fck;Cmax)γc=19.3MPa
(30)fctd=αctfctk,0.05γc=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 :

(31)σs=432.71+952.38εS454MPa
(32)σs=433.20+727.27εS466MPa
(33)σs=432.84+895.52εS493MPa

National annex provide ratio between ultimate and characteristic steel strain:

National annexεud/εuk
French0.9
Belgian0.8

For instance, according to the French National Annex:

Ductilityεudεuk
A22.5 ‰25 ‰
B45 ‰50 ‰
C67.5 ‰75 ‰

 

The ultimate strain of the elastic zone (εe) depends in any case on the considered combination. In the case of an ULS combination (persistent or transient):

(34)fyd=fykγs=500MPa1.15=435MPa
(35)Es=200000MPa
(36)εe=fydEs=435MPa200000MPa=2.175

 

Safety factors

 

Allowable stresses

(37)σs,adm=fyd=fyk/γs

Notes

Stresses

SymbolUnitDescription
fckMPaPermissible compressive stress of concrete (characteristic value at 28 days)
CmaxMPaCharacteristic compressive strength during processing
k1-Empirical coefficient, depends on the method of concrete pouring in the ground
k2-Empirical coefficient, depends on the concrete casting difficulties related to the geometry of the structure
k3-Empirical coefficient, depends on whether integrity checks are carried out
fckMPaCharacteristic compressive strength considered in the calculation of concrete walls
fcmMPaAverage compressive strength
fctmMPaAverage tensile strength
fct;k;0.05MPaCharacteristic tensile strength (5%)
fcdMPaPermissible compressive stress of concrete (design value)
αcc-Coefficient taking into account the long-term effects on the compressive strength of concrete
> French national annex = 1.0
> Belgian national annex = 0.85
αct-Coefficient taking into account the long-term effects on the tensile strength of concrete
> French national annex = 1.0
> Belgian national annex = 1.0
fydMPaPermissible stress of steel (design value)
fykMPaPermissible stress of steel (characteristic value)
γc-Partial coefficient on concrete strength

Strains

SymbolUnitDescription
εs1-Strain of the tensioned steel section
εs2-Strain of the compressed steel section
εc-Concrete strain
εe-Maximum elastic strain of steel

 

References