Plastic Analysis

This example demonstrates how to perform a plastic analysis in sectionproperties.

Create Geometry and Section

A monosymmetric I-section will be analysed, first we construct the geometry of the 200 mm deep section.

from sectionproperties.pre.library import mono_i_section


geom = mono_i_section(d=200, b_t=50, b_b=100, t_ft=12, t_fb=8, t_w=6, r=8, n_r=8)

Like geometric analyses, plastic analyses in sectionproperties are mesh independent.

from sectionproperties.analysis import Section


geom.create_mesh(mesh_sizes=0)
sec = Section(geometry=geom)
sec.plot_mesh(materials=False)

<Axes: title={'center': 'Finite Element Mesh'}>

Perfom Plastic Analysis

A plastic analysis first requires a geometric analysis to be performed. Let’s see what happens if we try to first run a plastic analysis.

sec.calculate_plastic_properties()

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
Cell In[3], line 1
----> 1 sec.calculate_plastic_properties()

File ~/checkouts/readthedocs.org/user_builds/sectionproperties/envs/latest/lib/python3.10/site-packages/sectionproperties/analysis/section.py:1120, in Section.calculate_plastic_properties(self, verbose)
   1118 if self.section_props.cx is None:
   1119     err = "Calculate geometric properties before performing a plastic analysis."
-> 1120     raise RuntimeError(err)
   1122 # check if any geometry are overlapped
   1123 if isinstance(self.geometry, sp_geom.CompoundGeometry):

RuntimeError: Calculate geometric properties before performing a plastic analysis.

Now we run the geometric analysis first.

sec.calculate_geometric_properties()
sec.calculate_plastic_properties()

Plastic Analysis Results

We can visualise the plastic centroid by calling the plot_centroids() method.

sec.plot_centroids()

<Axes: title={'center': 'Centroids'}>

sectionproperties will calculate the plastic section moduli of the cross-section during a plastic analysis. Assuming this section is made from steel (\(f_y = 250\) MPa) and there are no local instabilities, we can compare the yield and plastic capacity of the cross-section.

fy = 250  # yield stress in MPa

# calculate yield moment for the top & bottom flanges
my_t = fy * sec.get_z()[0]
my_b = fy * sec.get_z()[1]

# calculate plastic moment about x-axis
mp = fy * sec.get_s()[0]

# print results
print(f"My_t = {my_t / 1e6:.1f} kN.m")
print(f"My_b = {my_b / 1e6:.1f} kN.m")
print(f"Mp = {mp / 1e6:.1f} kN.m")

My_t = 36.4 kN.m
My_b = 43.5 kN.m
Mp = 46.2 kN.m

Thus, the section will yield about it’s top flange first at 36.4 kN.m. The plastic capacity of the section is reached at 46.2 kN.m (i.e. the entire cross-section is at yield stress).

The shape factors, i.e. the ratio between the plastic and elastic section moduli, can also be obtained.

print(f"SF_t = {sec.get_sf()[0]:.2f}")
print(f"SF_b = {sec.get_sf()[1]:.2f}")

SF_t = 1.27
SF_b = 1.06

Principal Axis Bending

Plastic section analysis about principal bending axes are also computed. As angle section is analysed to demonstrate the difference between global and principal axis bending.

from sectionproperties.pre.library import angle_section


geom = angle_section(d=150, b=90, t=12, r_r=10, r_t=5, n_r=8)
geom.create_mesh(mesh_sizes=0)
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
sec.calculate_plastic_properties()
sec.plot_centroids()

<Axes: title={'center': 'Centroids'}>

print(f"Sxx = {sec.get_s()[0]:.3e} mm3")
print(f"S11 = {sec.get_sp()[0]:.3e} mm3")
print(f"Syy = {sec.get_s()[1]:.3e} mm3")
print(f"S22 = {sec.get_sp()[1]:.3e} mm3")

Sxx = 1.135e+05 mm3
S11 = 1.210e+05 mm3
Syy = 4.572e+04 mm3
S22 = 4.376e+04 mm3

As expected, the plastic section moduli about the 11-principal axis is larger than any about the global axes.