Eurocode 2 model for concrete creep and shrinkage - EC2CreepMat

Implementation of aging viscoelastic model for concrete creep according to Eurocode 2 for concrete structures The model parameters are summarized in Tab. 34.

According to EC2, the compliance function is defined using the creep coefficient $\varphi$ as

$$J(t,t') = \frac{1}{E(t')} + \frac{\varphi(t,t')}{ 1.05 E_{cm}}$$ (92)

where $E_{cm}$ is the mean elastic modulus at the age of 28 days and $E(t')$ is the elastic modulus at the age of loading, $t'$.

Current implementation supports only linear creep which is valid only for stresses below 0.45 of the characteristic compressive strength at the time of loading.

The elastic modulus at age $t$ (in days) is defined as

$$E(t) = \left[ \exp \left( s \left(1-\sqrt{28/t} \right) \right) \right]^{0.3} E_{cm}$$ (93)

where $s$ is a cement-type dependent constant (0.2 for class R, 0.25 for class N and finally 0.38 for type S), and the mean secant elastic modulus at 28 days can be estimated from the mean compressive strength

$$E_{cm} = 22 \: \left(0.1 f_{cm,28} \right)^{0.3}$$ (94)

where $f_{cm,28}$ is in MPa and the resulting modulus is in GPa.

The creep coefficient is given by the expression from Annex B of the standard.

$$\varphi(t,t') = \varphi_{RH} \times \frac{16.8}{f_{cm}} \times \frac{1}{0.1 + t'^{0.2}} \left( \frac{t-t'}{\beta_H + t - t'_T} \right)^{0.3}$$ (95)

with

$$\varphi_{RH} = \left( 1 + \alpha_1 \frac{1-h_{env}}{0.1 \: (h_0)^{1/3}} \right) \times \alpha_2$$ (96)

$$\beta_H = 1.5 \left(1 + \left( 1.2 h_{env} \right)^{18} \right) h_0 + 250 \: \alpha_3 \leq 1500 \: \alpha_3$$ (97)

and

$$\alpha_1 = \alpha_2 = \alpha_3 = 1 \quad \mathrm{for} \: f_{cm} \leq 35 \: \mathrm{MPa}$$ (98)

else

$$\alpha_1 = \left( \frac{35}{f_{cm}} \right)^{0.7}$$ (99)

$$\alpha_2 = \left( \frac{35}{f_{cm}} \right)^{0.2}$$ (100)

$$\alpha_3 = \left( \frac{35}{f_{cm}} \right)^{0.5}$$ (101)

and $t'_T$ is the temperature-adusted age according to B.9 in the code. It is computed automatically if string temperatureDependent appears in the input record.

The shrinkage deformation is additively split into two parts: drying shrinkage $\varepsilon_{sh,d}$ and autogenous shrinkage $\varepsilon_{sh,a}$. Drying shrinkage strain at time $t$ is computed from

$$\varepsilon_{sh,d} = \frac{t-t_0}{t-t_0 + 0.04 \: h^{3/2}} \: k_h \: \varepsilon_{sh,d,0}$$ (102)

where $t_0$ is the duration of curing, $k_h$ is a thickness-dependent parameter and

$$\varepsilon_{sh,d,0} = 1.3175 \times 10^{-6} \left[ (220 + 110 \a... ...p \left( -0.1 \alpha_{ds2} \: f_{cm} \right) \right] \left( 1- h_{env}^3\right)$$ (103)

with $\alpha_{ds1} = 3$ for cement class S, 4 for class N, and 6 for class R, and $\alpha_{ds2} = 0.13$ for cement class S, 0.12 for class N, and 0.11 for class R.

Autogenous shrinkage strain can be comptued as

$$\varepsilon_{sh,a} = 2.5 \: (f_{cm} - 18) \left[ 1- \exp \left( -0.2 \sqrt{t} \right) \right] \times 10^{-6}$$ (104)

Table 34: EC2Creep material model - summary.

Description	EC2CreepMat model for concrete creep and shrinkage
Record Format	EC2CreepMat n(rn) # [ begOfTimeOfInterest(rn) #] [ endOfTimeOfInterest(rn) #] relMatAge(rn) # [ timeFactor(rn) #] stiffnessFactor(rn) # [ tAlpha(rn) #] fcm28(rn) # t0(rn) # cemType(in) # [ henv(rn) #] h0(rn) # shType(in) # [ spectrum ] [ temperatureDependent ]
Parameters	- num material model number
	- n Poisson's ratio
	- begOfTimeOfInterest determines the shortest time which is reasonably well captured by the approximated compliance function (default value is 0.1); the units are the time units of the analysis
	- endOfTimeOfInterest determines the longest time which is reasonably well captured by the approximated compliance function (if not provided it is read from the engineering model); the units are the time units of the analysis
	- relMatAge time shift used to specify the age of material on the begging of the analysis, the meaning is the material age at time t = 0;
	- timeFactor scaling factor transforming the actual time into appropriate units needed by the formulae of the eurocode. For analysis in days timeFactor = 1, for analysis in seconds timeFactor = 86,400.
	- stiffnessFactor scaling factor transforming predicted stiffness into appropriate units of the analysis, for analysis in MPa stiffnessFactor = 1.e6 (default), for Pa stiffnessFactor = 1
	- fcm28 mean compressive strength measured on cylinders at the age of 28 days in MPa
	- t0 duration of curing [day] (this is relevant only for drying shrinkage, not for creep)
	- cemType type of cement, 1 = class R, 2 = class N, 3 = class S
	- henv ambient relative humidity expressed as decimal
	- h0 effective member thickness in [mm] calculated according to EC2 as 2$\times$A/u where A is the cross-section are and u is the cross-section perimeter exposed to drying
	- shType shrinkage type; 0 = no shrinkage, 1 = both drying and autogenous shrinkage, 2 = drying shrinkage only, 3 = autogenous shrinkage only
	- spectrum this string switches on evaluation of the moduli of the aging Kelvin chain using the retardation spectrum of the compliance function, otherwise (default option) the least-squares method is used
	- temperatureDependent turns on the influence of temperature on concrete maturity (equivalent age concept) by default this option is not activated.
Supported modes	3dMat, PlaneStress, PlaneStrain, 1dMat, 2dPlateLayer,2dBeamLayer, 3dShellLayer