Biaxial Bending Diagram

This example demonstrates how to generate biaxial bending diagrams. We start by importing the necessary modules.

import numpy as np
from sectionproperties.pre.library import rectangular_section

from concreteproperties import (
    Concrete,
    ConcreteLinear,
    ConcreteSection,
    RectangularStressBlock,
    SteelBar,
    SteelElasticPlastic,
    add_bar_rectangular_array,
)
from concreteproperties.results import BiaxialBendingResults

Assign Materials

The materials used in this example will be 40 MPa concrete and 500 MPa steel, specified in accordance with AS 3600:2018.

concrete = Concrete(
    name="40 MPa Concrete",
    density=2.4e-6,
    stress_strain_profile=ConcreteLinear(elastic_modulus=32.8e3),
    ultimate_stress_strain_profile=RectangularStressBlock(
        compressive_strength=40,
        alpha=0.79,
        gamma=0.87,
        ultimate_strain=0.003,
    ),
    flexural_tensile_strength=3.8,
    colour="lightgrey",
)

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

Create Geometry and Concrete Section

In this example we’ll use the geometry from the Moment Curvature Analysis example, without the circular void.

col = rectangular_section(d=350, b=600, material=concrete)

# add bars to column
geom = add_bar_rectangular_array(
    geometry=col,
    area=450,
    material=steel,
    n_x=6,
    x_s=492 / 5,
    n_y=3,
    y_s=121,
    anchor=(54, 54),
    exterior_only=True,
)

conc_sec = ConcreteSection(geom)
conc_sec.plot_section()

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

Biaxial Bending Diagram

We can create a biaxial bending diagram by calling the biaxial_bending_diagram() method. We first create a biaxial bending diagram for pure bending, i.e. no axial force.

bb_res = conc_sec.biaxial_bending_diagram(n_points=24, progress_bar=False)

We can plot the biaxial bending diagram by calling the plot_diagram() method.

bb_res.plot_diagram()

<Axes: title={'center': 'Biaxial Bending Diagram, $N = 0.000e+00$'}, xlabel='Bending Moment $M_x$', ylabel='Bending Moment $M_y$'>

Say we know the column has an axial force of 1000 kN, we can generate a biaxial bending diagram for this case by passing this to the parameter n.

bb_res = conc_sec.biaxial_bending_diagram(n=1000e3, n_points=24, progress_bar=False)
bb_res.plot_diagram()

<Axes: title={'center': 'Biaxial Bending Diagram, $N = 1.000e+06$'}, xlabel='Bending Moment $M_x$', ylabel='Bending Moment $M_y$'>

Plotting Multiple Diagrams

We can also plot multiple diagrams at once by using the BiaxialBendingResults.plot_multiple_diagrams() class method. Here we will generate biaxial bending diagrams with varying axial load from pure bending up to the decompression point. We start by examining the moment interaction diagrams to find the decompression point.

mi_x = conc_sec.moment_interaction_diagram(progress_bar=False)
mi_y = conc_sec.moment_interaction_diagram(theta=np.pi / 2, progress_bar=False)
mi_x.plot_diagram()
mi_y.plot_diagram(moment="m_y")
print(f"Decompression point for M_x is N = {mi_x.results[1].n / 1e3:.2f} kN")
print(f"Decompression point for M_y is N = {mi_y.results[1].n / 1e3:.2f} kN")

Decompression point for M_x is N = 7452.19 kN
Decompression point for M_y is N = 7444.80 kN

Let’s generate 5 biaxial bending curves from pure bending (\(N=0\) kN) to close to the decompression point (\(N=7400\) kN).

n_list = np.linspace(0, 7400e3, 5)
biaxial_results = []

for n in n_list:
    biaxial_results.append(
        conc_sec.biaxial_bending_diagram(n=n, n_points=24, progress_bar=False)
    )

Now that we have the results, we can plot all the diagrams on a 3D plot!

BiaxialBendingResults.plot_multiple_diagrams_3d(biaxial_results)

<Axes3D: xlabel='Bending Moment $M_x$', ylabel='Bending Moment $M_y$', zlabel='Axial Force $N$'>

We can also generate a 2D plot of the biaxial bending results.

BiaxialBendingResults.plot_multiple_diagrams_2d(biaxial_results, fmt="o-")

<Axes: title={'center': 'Biaxial Bending Diagram'}, xlabel='Bending Moment $M_x$', ylabel='Bending Moment $M_y$'>

We can specify custom labels for the above plot.

labels = []

for bb_res in biaxial_results[::2]:
    labels.append(f"N = {bb_res.n / 1e3:.0f} kN")

ax = BiaxialBendingResults.plot_multiple_diagrams_2d(
    biaxial_results[::2], fmt="-", labels=labels, render=False
)
ax.set_xlabel("Bending Moment $M_x$ [kN.m]")
ax.set_ylabel("Bending Moment $M_y$ [kN.m]")
import matplotlib.pyplot as plt


plt.show()