m_n_kappa.material.ConcreteCompressionParabolaRectangle#

class m_n_kappa.material.ConcreteCompressionParabolaRectangle(f_cm, E_cm)#

Bases: ConcreteCompression

parabola-rectangle behaviour of concrete under compression according to EN 1992-1-1 [1]

New in version 0.1.0.

Parameters:

  • f_cm (float) – mean concrete cylinder compressive strength \(f_\mathrm{cm}\)

  • E_cm (float) – mean elasticity modulus of concrete \(E_\mathrm{cm}\)

See also

ConcreteCompressionNonlinear

Describes non-linear behaviour of concrete under compression

ConcreteCompressionBiLinear

describes bi-linear behaviour of concrete under compression

Notes

The formula for computation of the parabola-rectangle behaviour of concrete by EN 1992-1-1 [1], Formula 3.17 is given as follows. Formula (1) is valid in the range \(0 < |\varepsilon| < |\varepsilon_\mathrm{c}|\). The here given values are all absolute values. As this model applies to the compression-range all values must be multiplied by (-1).

(1)#\[ \begin{align}\begin{aligned}\sigma_\mathrm{c} & = f_\mathrm{c} \cdot \left[1 - \left(1 - \frac{\varepsilon}{\varepsilon_\mathrm{c}} \right)^{n} \right] & & \text{ for } 0 \leq \varepsilon \leq \varepsilon_\mathrm{c}\\\sigma_\mathrm{c} & = f_\mathrm{c} & & \text{ for } \varepsilon_\mathrm{c} \leq \varepsilon \leq \varepsilon_\mathrm{cu}\end{aligned}\end{align} \]

where

(2)#\[ \begin{align}\begin{aligned}\text{ for } f_\mathrm{ck} \leq 50 \text{ N/mm :sup:`2`}: &\\& \varepsilon_\mathrm{c}(Permil) = 2.0\\& \varepsilon_\mathrm{cu}(Permil) = 3.5\\& n = 2.0\\\text{for } f_\mathrm{ck} \geq 50 \text{ N/mm :sup:`2`} &\\& \varepsilon_\mathrm{c}(Permil) = 2.0 + 0.085 \cdot (f_\mathrm{ck} - 50)^{0.53}\\& \varepsilon_\mathrm{cu}(Permil) = 2.6 + 35 \left[\frac{90-f_\mathrm{ck}}{100} \right]^{4}\\& n = 1.4 + 23.4 \cdot \left[ \frac{90-f_\mathrm{ck}}{100} \right]\end{aligned}\end{align} \]

where \(\varepsilon_\mathrm{c}\) is the strain at peak stress and \(\varepsilon_\mathrm{cu}\) is the strain at failure.

../_images/material_concrete_parabola_rectangle-light.svg
../_images/material_concrete_parabola_rectangle-dark.svg

Parabola-rectangle relationship of concrete by EN 1992-1-1 [1], Fig. 3.3#

References

Examples

The stress-strain relationship of concrete under compression is computed as follows.

>>> from m_n_kappa.material import ConcreteCompressionParabolaRectangle
>>> f_cm = 30.0 # mean concrete compressive strength
>>> E_cm = 33000 # modulus of elasticity of concrete
>>> concrete = ConcreteCompressionParabolaRectangle(f_cm=f_cm, E_cm=E_cm)
>>> concrete.stress_strain()
[[-9.625, -0.0005], [-16.5, -0.001], [-20.625, -0.0015], [-22.0, -0.002], [-22.0, -0.0035]]

Methods

stress(strain)

computation of stresses according to formula (1)

stress_strain()

stress-strain points of the material

Attributes

E_cm

mean elasticity modulus of concrete \(E_\mathrm{cm}\)

c

strain at peak stress \(\varepsilon_\mathrm{c}\) (see Formula (2))

cu

failure strain of concrete \(\varepsilon_\mathrm{cu}\) (see Formula (2))

f_ck

characteristic concrete cylinder compressive strength \(f_\mathrm{ck}\)

f_cm

mean concrete cylinder compressive strength \(f_\mathrm{cm}\)

n

exponent in formula (1) given in formula (2)

strains

Strain-values where stresses are computed.

yield_strain

strain up to which the concrete is assumed to be linear-elastic \(\varepsilon_\mathrm{y}\)
stress(strain)#

computation of stresses according to formula (1)

Parameters:

strain (float) – strain to compute corresponding stress

Returns:

stress to the given strain

Return type:

float

stress_strain()#

stress-strain points of the material

Return type:

list

property E_cm: float#

mean elasticity modulus of concrete \(E_\mathrm{cm}\)

property c: float#

strain at peak stress \(\varepsilon_\mathrm{c}\) (see Formula (2))

property cu: float#

failure strain of concrete \(\varepsilon_\mathrm{cu}\) (see Formula (2))

property f_ck: float#

characteristic concrete cylinder compressive strength \(f_\mathrm{ck}\)

property f_cm: float#

mean concrete cylinder compressive strength \(f_\mathrm{cm}\)

property n: float#

exponent in formula (1) given in formula (2)

property strains: list#

Strain-values where stresses are computed.

Current strain-values are:

  • \(0.25 \cdot \varepsilon_\mathrm{c}\)

  • \(0.50 \cdot \varepsilon_\mathrm{c}\)

  • \(0.75 \cdot \varepsilon_\mathrm{c}\)

  • \(\varepsilon_\mathrm{c}\)

  • \(\varepsilon_\mathrm{cu}\)

property yield_strain: float#

strain up to which the concrete is assumed to be linear-elastic \(\varepsilon_\mathrm{y}\)