Composite Analysis#
This example highlights how material properties allow composite analysis in sectionproperties.
Geometric vs. Composite#
The default analysis type in sectionproperties is purely geometric, i.e. cross-section properties are reported based on the geometry only. In this analysis, all geometries are assigned the default material:
[1]:
from sectionproperties.analysis import Section
from sectionproperties.pre.library import rectangular_section
rect_geom = rectangular_section(d=100, b=50)
rect_geom.material
[1]:
Material(name='default', elastic_modulus=1, poissons_ratio=0, yield_strength=1, density=1, color='w')
The default material has a unit elastic modulus, yield strength and density, and a Poisson’s ratio of zero.
Geometric-only analysis allows geometric properties to be obtained:
[2]:
rect_geom.create_mesh(mesh_sizes=10) # create mesh
rect_sec = Section(geometry=rect_geom)
rect_sec.calculate_geometric_properties()
ixx, iyy, ixy = rect_sec.get_ic() # get second moments of area
print(f"Ixx = {ixx:.5e} mm4")
Ixx = 4.16667e+06 mm4
When a material property gets added to a geometry, the analysis becomes composite.
[3]:
from sectionproperties.pre import Material
# assign steel to the geometry
steel = Material(
name="Steel",
elastic_modulus=200e3,
poissons_ratio=0.3,
density=7.85e-6,
yield_strength=500,
color="grey",
)
rect_geom.material = steel
# recreate mesh and section
rect_geom.create_mesh(mesh_sizes=10)
rect_sec = Section(geometry=rect_geom)
rect_sec.calculate_geometric_properties()
Cross-section properties are now modulus weighted as the assumption is that there are multiple regions with multiple different material properties. We can no longer obtain geometric-only properties:
[4]:
ixx, iyy, ixy = rect_sec.get_ic() # get second moments of area
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call last)
Cell In[4], line 1
----> 1 ixx, iyy, ixy = rect_sec.get_ic() # get second moments of area
File ~/checkouts/readthedocs.org/user_builds/sectionproperties/envs/latest/lib/python3.10/site-packages/sectionproperties/analysis/section.py:2081, in Section.get_ic(self)
2079 msg = "Attempting to get a geometric only property for a composite analysis"
2080 msg += " (material properties have been applied). Consider using get_eic()."
-> 2081 raise RuntimeError(msg)
2083 try:
2084 assert self.section_props.ixx_c is not None
RuntimeError: Attempting to get a geometric only property for a composite analysis (material properties have been applied). Consider using get_eic().
In this case, we need to get the modulus weighted second moments of area. Note we can still extract geometric properties by using a reference elastic modulus.
[5]:
# get modulus weighted second moments of area
eixx, eiyy, eixy = rect_sec.get_eic()
print(f"E.Ixx = {eixx:.5e} N.mm2")
# use reference elastic modulus to get transformed properties
ixx, iyy, ixy = rect_sec.get_eic(e_ref=steel)
print(f"Ixx = {ixx:.5e} mm4")
E.Ixx = 8.33333e+11 N.mm2
Ixx = 4.16667e+06 mm4
Steel-Timber Composite Section#
The following section models a composite timber floor and steel beam section. A universal steel beam (310UB40.4) is modelled with a 100D x 600W timber panel placed on its top flange.
Create the Materials#
[6]:
# create the steel material
steel = Material(
name="Steel",
elastic_modulus=200e3,
poissons_ratio=0.3,
density=7.85e-6,
yield_strength=500,
color="grey",
)
# create the timber material
timber = Material(
name="Timber",
elastic_modulus=8e3,
poissons_ratio=0.35,
yield_strength=20,
density=0.78e-6,
color="burlywood",
)
Create the Geometry#
[7]:
from sectionproperties.pre.library import i_section
# universal steel beam
ub = i_section(d=304, b=165, t_f=10.2, t_w=6.1, r=11.4, n_r=8, material=steel)
# timber floor panel
panel = rectangular_section(d=100, b=600, material=timber)
panel = panel.align_center(align_to=ub).align_to(other=ub, on="top")
# combine geometry
geom = ub + panel
Create Mesh and Section Object#
[8]:
# 10 mm2 mesh for UB, 500 mm2 mesh for timber
geom.create_mesh(mesh_sizes=[10, 500])
sec = Section(geometry=geom)
sec.plot_mesh()
[8]:
<Axes: title={'center': 'Finite Element Mesh'}>
Perform Analysis#
[9]:
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
sec.calculate_plastic_properties()
Display Analysis Results#
We can plot the various centroids found by the analyses.
[10]:
sec.plot_centroids()
[10]:
<Axes: title={'center': 'Centroids'}>
We can also print all the calculated section properties to the terminal, note that because have conducted a composite analysis, modulus weighted properties are displayed.
[11]:
sec.display_results()
Section Properties ┏━━━━━━━━━━━┳━━━━━━━━━━━━━━━┓ ┃ Property ┃ Value ┃ ┡━━━━━━━━━━━╇━━━━━━━━━━━━━━━┩ │ area │ 6.521094e+04 │ │ perimeter │ 2.306078e+03 │ │ mass │ 8.770585e-02 │ │ e.a │ 1.522187e+09 │ │ e.qx │ 3.283325e+11 │ │ e.qy │ 1.255805e+11 │ │ e.ixx_g │ 1.019283e+14 │ │ e.iyy_g │ 2.628963e+13 │ │ e.ixy_g │ 2.708743e+13 │ │ cx │ 8.250000e+01 │ │ cy │ 2.156978e+02 │ │ e.ixx_c │ 3.110771e+13 │ │ e.iyy_c │ 1.592924e+13 │ │ e.ixy_c │ -9.375000e-02 │ │ e.zxx+ │ 1.652010e+11 │ │ e.zxx- │ 1.442190e+11 │ │ e.zyy+ │ 5.309747e+10 │ │ e.zyy- │ 5.309747e+10 │ │ rx │ 1.429552e+02 │ │ ry │ 1.022971e+02 │ │ e.i11_c │ 3.110771e+13 │ │ e.i22_c │ 1.592924e+13 │ │ phi │ 0.000000e+00 │ │ e.z11+ │ 1.652010e+11 │ │ e.z11- │ 1.442190e+11 │ │ e.z22+ │ 5.309747e+10 │ │ e.z22- │ 5.309747e+10 │ │ r11 │ 1.429552e+02 │ │ r22 │ 1.022971e+02 │ │ e_eff │ 2.334252e+04 │ │ g_eff │ 8.873037e+03 │ │ nu_eff │ 3.153622e-01 │ │ e.j │ 2.169153e+12 │ │ x_se │ 8.255462e+01 │ │ y_se │ 3.200131e+02 │ │ x1_se │ 5.461533e-02 │ │ y2_se │ 1.043152e+02 │ │ x_st │ 8.255367e+01 │ │ y_st │ 3.179106e+02 │ │ e.gamma │ 8.613474e+16 │ │ e.a_sx │ 6.005132e+08 │ │ e.a_sy │ 4.252562e+08 │ │ e.a_s11 │ 6.005132e+08 │ │ e.a_s22 │ 4.252562e+08 │ │ beta_x+ │ 2.165060e+02 │ │ beta_x- │ -2.165060e+02 │ │ beta_y+ │ 1.092307e-01 │ │ beta_y- │ -1.092307e-01 │ │ beta_11+ │ 2.165060e+02 │ │ beta_11- │ -2.165060e+02 │ │ beta_22+ │ 1.092307e-01 │ │ beta_22- │ -1.092307e-01 │ │ x_pc │ 8.250000e+01 │ │ y_pc │ 2.954820e+02 │ │ x11_pc │ 8.250000e+01 │ │ y22_pc │ 2.954820e+02 │ │ mp_xx │ 4.858134e+08 │ │ mp_yy │ 5.335011e+08 │ │ mp_11 │ 4.858134e+08 │ │ mp_22 │ 5.335011e+08 │ └───────────┴───────────────┘
We can also get transformed properties by specifying a reference material.
[12]:
ixx_timber, _, _ = sec.get_eic(e_ref=timber)
ixx_steel, _, _ = sec.get_eic(e_ref=steel)
print(f"Ixx,t = {ixx_timber:.3e} mm4")
print(f"Ixx,s = {ixx_steel:.3e} mm4")
Ixx,t = 3.888e+09 mm4
Ixx,s = 1.555e+08 mm4
A plastic analysis for composite sections will calculate plastic moments rather than plastic section moduli. The plastic moment assumes all geometry fibres reach the yield strength.
[13]:
mp_xx, _ = sec.get_mp()
print(f"Mp = {mp_xx / 1e6:.1f} kN.m")
Mp = 485.8 kN.m
Stress Analysis#
[14]:
stress = sec.calculate_stress(n=-100e3, mxx=-120e6, vy=-75e3)
[15]:
stress.plot_stress(stress="m_zz")
[15]:
<Axes: title={'center': 'Stress Contour Plot - $\\sigma_{zz,\\Sigma M}$'}>
[16]:
stress.plot_stress(stress="vm")
[16]:
<Axes: title={'center': 'Stress Contour Plot - $\\sigma_{vM}$'}>
We can plot only a specific list of materials by including the material_list argument. In the above plot it is difficult to see the stress in the timber so we set material_list=[timber].
[17]:
stress.plot_stress(stress="vm", material_list=[timber])
[17]:
<Axes: title={'center': 'Stress Contour Plot - $\\sigma_{vM}$'}>