5. Moment Interaction Diagram

This example demonstrates how to generate moment interaction diagrams. We start by importing the necessary modules.

import numpy as np
from concreteproperties.material import Concrete, Steel
from concreteproperties.stress_strain_profile import (
    ConcreteLinear,
    RectangularStressBlock,
    SteelElasticPlastic,
)
from sectionproperties.pre.library.concrete_sections import concrete_rectangular_section
from concreteproperties.concrete_section import ConcreteSection
from concreteproperties.results import MomentInteractionResults

5.1. Assign Materials

The materials used in this example will be 32 MPa concrete and 500 MPa steel, specified in accordance with AS 3600:2018.

concrete = Concrete(
    name="32 MPa Concrete",
    density=2.4e-6,
    stress_strain_profile=ConcreteLinear(elastic_modulus=30.1e3),
    ultimate_stress_strain_profile=RectangularStressBlock(
        compressive_strength=32,
        alpha=0.802,
        gamma=0.89,
        ultimate_strain=0.003,
    ),
    alpha_squash=0.85,
    flexural_tensile_strength=3.4,
    colour="lightgrey",
)

steel = Steel(
    name="500 MPa Steel",
    density=7.85e-6,
    stress_strain_profile=SteelElasticPlastic(
        yield_strength=500,
        elastic_modulus=200e3,
        fracture_strain=0.05,
    ),
    colour="grey",
)

5.2. Create Geometry and Concrete Section

The geometry used in this example is identical to that used in Calculating Area Properties.

geom = concrete_rectangular_section(
    b=400,
    d=600,
    dia_top=20,
    n_top=3,
    dia_bot=24,
    n_bot=3,
    n_circle=4,
    cover=30,
    area_top=310,
    area_bot=450,
    conc_mat=concrete,
    steel_mat=steel,
)

conc_sec = ConcreteSection(geom)
conc_sec.plot_section()

<AxesSubplot:title={'center':'Reinforced Concrete Section'}>

5.3. Moment Interaction Diagram

We perform a moment interaction analysis by calling the moment_interaction_diagram() method.

mi_res = conc_sec.moment_interaction_diagram()

We can plot the moment interaction diagram by calling the plot_diagram() method.

mi_res.plot_diagram()

<AxesSubplot:title={'center':'Moment Interaction Diagram'}, xlabel='Bending Moment', ylabel='Axial Force'>

What if we knew that we had equal bending moments about the x and y axis? In this case the neutral axis angle would be \(\theta = \pi / 2\). Let’s generate a moment interaction diagram for this case.

mi_res = conc_sec.moment_interaction_diagram(theta=np.pi / 2)
mi_res.plot_diagram()

<AxesSubplot:title={'center':'Moment Interaction Diagram'}, xlabel='Bending Moment', ylabel='Axial Force'>

It’s also possible to plot a diagram for positive and negative bending (sagging and hogging). We do this by setting the parameter m_neg=True. We can control the number of points between the pure bending point and decompression point by modifying the parameter n_points.

mi_res = conc_sec.moment_interaction_diagram(m_neg=True, n_points=48)
mi_res.plot_diagram()

<AxesSubplot:title={'center':'Moment Interaction Diagram'}, xlabel='Bending Moment', ylabel='Axial Force'>

In the above plot the bending capacity is higher for positive bending and not symmetric. This is because the bottom bars (3N24) provide more area than the top bars (3N20).

5.4. Plotting Multiple Diagrams

We can also plot multiple diagrams at once by using the MomentInteractionResults.plot_multiple_diagrams() class method. In this example we’ll create four different reinforced concrete cross-sections with different reinforcement ratios.

# create lists to hold results and labels
mi_results = []
labels = []

# create four different sections with increasing reinforcement
# and peform a moment interaction analysis
for idx in range(4):
    geom = concrete_rectangular_section(
        b=400,
        d=600,
        dia_top=16,
        n_top=6,
        dia_bot=16,
        n_bot=6,
        n_circle=4,
        cover=66,
        area_top=200 * (idx + 1),
        area_bot=200 * (idx + 1),
        conc_mat=concrete,
        steel_mat=steel,
    )

    conc_sec = ConcreteSection(geom)
    mi_results.append(conc_sec.moment_interaction_diagram())
    labels.append("p = {0}".format(0.01 * (idx + 1)))

# plot all the diagrams on one image
MomentInteractionResults.plot_multiple_diagrams(
    moment_interaction_results=mi_results, labels=labels
)

<AxesSubplot:title={'center':'Moment Interaction Diagram'}, xlabel='Bending Moment', ylabel='Axial Force'>