Slim-Floor double T-profile

Units: Millimeter [mm], Newton [N]

Geometry and materials

The steel-part of this slim-floor consists of a bottom-flange where two webs are welded on top. Between these two webs a higher strength concrete is introduced that is reinforced by a top- and a bottom-layer.

Geometry: Cross-section of slim-floor beam with double T-steel profile

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

Material: Concrete non-linear stress-strain-relationship in compression acc. EN 1992-1-1 [1]

Material: Stress-strain-relationship of concrete in tension

Slim-floor beam with I-profile cross-section

>>> from m_n_kappa import Steel, Rectangle, Concrete, RebarLayer, Reinforcement
>>> concrete_core = Rectangle(top_edge=0.0, bottom_edge=200, width=300)
>>> concrete_material = Concrete(f_cm=60)
>>> concrete_slab = concrete_core + concrete_material
>>> left_web = Rectangle(
...     top_edge=40, bottom_edge=concrete_core.bottom_edge, width=8.0, right_edge=-0.5*concrete_core.left_edge)
>>> right_web = Rectangle(
...     top_edge=40, bottom_edge=concrete_core.bottom_edge, width=8.0, left_edge=0.5*concrete_core.left_edge)
>>> bottom_flange = Rectangle(
...     top_edge=concrete_core.bottom_edge, bottom_edge=concrete_core.bottom_edge + 12.0,
...     width=596.0)
>>> girder = left_web + right_web + bottom_flange
>>> steel = Steel(f_y=460, f_u=500, failure_strain=0.15)
>>> steel_girder = girder + steel
>>> reinforcement = Reinforcement(f_s=500, f_su=550, failure_strain=0.25)
>>> rebar_top_layer = RebarLayer(rebar_diameter=20, centroid_z=38.0, rebar_number=5, width=concrete_core.width - 2 * 28)
>>> rebar_bottom_layer = RebarLayer(
...     rebar_diameter=16.0, centroid_z=bottom_flange.bottom_edge-46.0,
...     rebar_number=3, width=rebar_top_layer.width)
>>> rebar_layers = rebar_top_layer + rebar_top_layer
>>> rebars = rebar_layers + reinforcement
>>> cross_section = concrete_slab + steel_girder + rebars

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

References

[1]

EN 1992-1-1: Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings, European Committee of Standardization (CEN), 2004ss_section_computation.rst