Symmetric steel profile

This template creates a HEB 200 steel-profile of steel-grade S355.

Units: Millimeter [mm], Newton [N]

The geometry-values are:

• b_fo=200: width of the top-flange \(b_\mathrm{fo} = 200\) mm (and the bottom-flange)

• t_fo=15: thickness of the top-flange \(t_\mathrm{fo} = 15\) mm (and the bottom-flange)

• h_w=200-2*15: height of the steel web \(h_\mathrm{w} = 200-2 \cdot 15 = 170\) mm

• t_w=9.5: thickness of the steel web \(t_\mathrm{w} = 9.5\) mm

The material-model is isotropic steel using following values:

• f_y=355: yield strength of the steel \(f_\mathrm{y} = 355\) N/mm²

• f_y=400: tensile strength of the steel \(f_\mathrm{u} = 400\) N/mm²

• failure_strain=0.15: failure strain of the steel \(\varepsilon_\mathrm{u} = 15\) %

>>> from m_n_kappa import IProfile, Steel
>>> i_profile = IProfile(top_edge=0.0, b_fo=200, t_fo=15, h_w=200-2*15, t_w=15, centroid_y=0.0)
>>> steel = Steel(f_y=355.0, f_u=400, failure_strain=0.15)
>>> cross_section = i_profile + steel

Geometry: Symmetric I-profile

Material: Bi-linear stress-strain-relationship with hardening of steel

Computation

The cross_section you created above is the basis to do a variety of computations:

Curvature

In case you want to compute a single curvature-value from a given strain at a given position, you first have to define strain and its position using StrainPosition strain_position is the boundary-condition, that is passed to MKappaByStrainPosition that is computing the curvature.

>>> from m_n_kappa import StrainPosition, MKappaByStrainPosition
>>> strain_position = StrainPosition(strain=-0.002, position=0.0, material="")
>>> computation = MKappaByStrainPosition(
...     cross_section=cross_section,
...     strain_position = strain_position,
...     positive_curvature=True)

After computation you can extract the results as follows:

• m_n_kappa.MKappaByStrainPosition.successful: if True equilibrium of horizontal forces has been achieved during computation

• m_n_kappa.MKappaByStrainPosition.axial_force: computed axial forces that should be near zero as this is what the computation is aimed at

• m_n_kappa.MKappaByStrainPosition.moment: computed moment

• m_n_kappa.MKappaByStrainPosition.curvature: computed curvature

• m_n_kappa.MKappaByStrainPosition.neutral_axis: vertical position of the neutral axis (strain \(\varepsilon=0\))

See also

Moment-Curvature-Curve: further explanations regarding computation of a single moment-curvature-point

M-\(\kappa\)-curve

The \(M\)-\(\kappa\)-curve is easily computed by passing the created cross_section to MKappaCurve. You only have to decide if you want only the positive moment-curvature-points, the negative moment-curvature-points or both.

>>> from m_n_kappa import MKappaCurve
>>> positive_m_kappa = MKappaCurve(cross_section=cross_section)
>>> negative_m_kappa = MKappaCurve(
...     cross_section=cross_section,
...     include_positive_curvature=False,
...     include_negative_curvature=True)
>>> full_m_kappa = MKappaCurve(
...     cross_section=cross_section,
...     include_positive_curvature=True,
...     include_negative_curvature=True)

The computed points are then stored in the attribute m_kappa_points that returns MKappaCurvePoints-object.

See also

Moment-Curvature-Curve : further explanation regarding computation of the Moment- Curvature-Curve

Beam-Deformation

For computation of the \(M\)-\(\kappa\)-curves in a beam you need the loading-scenario beside your cross_section. And you should decide in how many elements the beam shall be split into (see element_number). In case you also want to consider the effective widths you may set consider_widths=True.

>>> from m_n_kappa import SingleSpanUniformLoad, Beam
>>> loading = SingleSpanUniformLoad(length=8000, load=1.0)
>>> beam = Beam(cross_section=cross_section, element_number=10, load=loading)
>>> beam_consider_widths = Beam(
...     cross_section=cross_section,
...     element_number=10,
...     load=loading,
...     consider_widths=True)

The computed beams allow you to do a number of analysis, like:

• m_n_kappa.Beam.deformation_over_beam_length(): computes the deformation at each node along the beam under the given load

• m_n_kappa.Beam.deformations(): computes the deformation at the given position for the relevant load-steps

• m_n_kappa.Beam.deformations_at_maximum_deformation_position(): same like m_n_kappa.Beam.deformations() but at the position of the beam where the maximum deformation occurred under the given loading.

See also

Loading: further explanation of loading scenarios

Deformation : further explanation regarding computation of beam-deformation