I-Profile slim-floor beam

Units: Millimeter [mm], Newton [N]

Geometry and materials

The code below creates a cross_section that represents a slim-floor-beam where a steel I-profile (HEB 200) is integrated into a solid concrete-slab of type C30/35.

The concrete_slab is split into five parts:

• concrete-slab left and right of the steel-beam

• concrete within each of the two chambers

• concrete-cover above the steel-profile

Each of these parts is an individual Rectangle. The overall bottom-edge of the concrete-slab is similar to the bottom-edge of the I-profile. The non-linear relationship is used to model the stress-strain-relationship of the used Concrete.

Also integrated into the concrete-slab are ``rebar_layer``s, consisting of a number of reinforcement bars. The bi-linear stress-strain-relationship with hardening is used for the reinforcement-bars as well as for the I-profile (see Reinforcement, Steel).

Geometry: Cross-section of slim-floor beam with I-profile and solid slab

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 IProfile, Steel, Rectangle, Concrete, RebarLayer, Reinforcement
>>> concrete_slab_left = Rectangle(top_edge=0.0, bottom_edge=240, width=0.5*(2000-200), left_edge=-1000)
>>> concrete_slab_right = Rectangle(top_edge=0.0, bottom_edge=240, width=0.5*(2000-200), right_edge=1000)
>>> concrete_top_cover = Rectangle(top_edge=0.0, bottom_edge=50, width=200)
>>> concrete_left_chamber = Rectangle(top_edge=50+15.4, bottom_edge=50+200-15.4, width=(200-9.5)/2, left_edge=-200/2)
>>> concrete_right_chamber = Rectangle(top_edge=50+15.4, bottom_edge=50+200-15.4, width=(200-9.5)/2, right_edge=200/2)
>>> concrete_slab = concrete_slab_left + concrete_slab_right + concrete_top_cover + concrete_left_chamber + concrete_right_chamber
>>> concrete = Concrete(f_cm=30+8, )
>>> concrete_section = concrete_slab + concrete
>>> reinforcement = Reinforcement(f_s=500, f_su=550, failure_strain=0.15)
>>> top_layer = RebarLayer(
...     centroid_z=0.5*50, width=2000, rebar_horizontal_distance=200, rebar_diameter=10)
>>> top_rebar_layer = reinforcement + top_layer
>>> bottom_layer_left = RebarLayer(
...     centroid_z=240-10, width=2000, rebar_horizontal_distance=100, rebar_diameter=10, left_edge=-2000/2+10)
>>> bottom_layer_right = RebarLayer(
...     centroid_z=240-10, width=(2000-200-2*10)/2, rebar_horizontal_distance=100, rebar_diameter=10, right_edge=2000/2+10)
>>> bottom_rebar_layer_left = reinforcement + bottom_layer_left
>>> bottom_rebar_layer_right = reinforcement + bottom_layer_right
>>> rebar_layer = top_rebar_layer + bottom_rebar_layer_left + bottom_rebar_layer_right
>>> i_profile = IProfile(
...     top_edge=100.0, 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)
>>> steel_section = i_profile + steel
>>> cross_section = concrete_section + rebar_layer + steel_section

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), 2004