8. Composite Sections#
This example demonstrates how to concreteproperties can be used to analyse composite sections. We start by importing the necessary modules.
[1]:
import numpy as np
from concreteproperties.material import Concrete, Steel, SteelBar
import concreteproperties.stress_strain_profile as ssp
import sectionproperties.pre.library.primitive_sections as sp_ps
import sectionproperties.pre.library.steel_sections as sp_ss
import sectionproperties.pre.library.concrete_sections as sp_cs
import concreteproperties.pre as cp_pre
from concreteproperties.concrete_section import ConcreteSection
import concreteproperties.results as res
8.1. Assign Materials#
The materials used in this example will be 50 MPa concrete with 500 MPa reinforcing steel and 300 MPa structural steel.
[2]:
concrete = Concrete(
name="50 MPa Concrete",
density=2.4e-6,
stress_strain_profile=ssp.ConcreteLinearNoTension(
elastic_modulus=34.8e3,
ultimate_strain=0.003,
compressive_strength=0.9 * 50,
),
ultimate_stress_strain_profile=ssp.RectangularStressBlock(
compressive_strength=50,
alpha=0.775,
gamma=0.845,
ultimate_strain=0.003,
),
flexural_tensile_strength=4.2,
colour="lightgrey",
)
steel_300 = Steel(
name="300 MPa Structural Steel",
density=7.85e-6,
stress_strain_profile=ssp.SteelElasticPlastic(
yield_strength=300,
elastic_modulus=200e3,
fracture_strain=0.05,
),
colour="tan",
)
steel_bar = SteelBar(
name="500 MPa Steel Bar",
density=7.85e-6,
stress_strain_profile=ssp.SteelElasticPlastic(
yield_strength=500,
elastic_modulus=200e3,
fracture_strain=0.05,
),
colour="grey",
)
8.2. Square Column with Universal Beam#
First we will analyse a 500 mm x 500 mm square column with 12N20 bars and a central 310UC97 steel column.
[3]:
# create 500 square concrete
conc = sp_ps.rectangular_section(d=500, b=500, material=concrete)
# create 310UC97 and centre to column
uc = sp_ss.i_section(
d=308,
b=305,
t_f=15.4,
t_w=9.9,
r=16.5,
n_r=8,
material=steel_300,
).align_center(align_to=conc)
# cut hole in concrete for UC then add UC
geom = conc - uc + uc
# add 12N20 reinforcing bars
geom = cp_pre.add_bar_rectangular_array(
geometry=geom,
area=310,
material=steel_bar,
n_x=4,
x_s=132,
n_y=4,
y_s=132,
anchor=(52, 52),
exterior_only=True,
)
# create concrete section and plot
conc_sec = ConcreteSection(geom)
conc_sec.plot_section()
[3]:
<AxesSubplot:title={'center':'Reinforced Concrete Section'}>
8.2.1. Elastic Stress#
Calculate the elastic stress under a bending moment of 100 kN.m.
[4]:
el_stress = conc_sec.calculate_uncracked_stress(m_x=100e6)
el_stress.plot_stress()
[4]:
<AxesSubplot:title={'center':'Stress'}>
8.2.2. Cracked Stress#
Calculate the cracked stress under a bending moment of 500 kN.m.
[5]:
cr_res = conc_sec.calculate_cracked_properties()
cr_stress = conc_sec.calculate_cracked_stress(cracked_results=cr_res, m=500e6)
cr_stress.plot_stress()
[5]:
<AxesSubplot:title={'center':'Stress'}>
8.2.3. Moment Curvature Diagram#
Generate a moment-curvature diagram and plot the stress after yielding of the steel.
[6]:
mk_res = conc_sec.moment_curvature_analysis()
[7]:
mk_res.plot_results(fmt="kx-")
[7]:
<AxesSubplot:title={'center':'Moment-Curvature'}, xlabel='Curvature', ylabel='Moment'>
[8]:
serv_stress = conc_sec.calculate_service_stress(
moment_curvature_results=mk_res, m=None, kappa=2e-5
)
serv_stress.plot_stress()
[8]:
<AxesSubplot:title={'center':'Stress'}>
8.2.4. Ultimate Results#
Finally the ultimate bending capacity about the x and y axes will be calculated, as well as generating an interaction diagram, a biaxial bending diagram and visualising the ultimate stresses.
[9]:
ult_res_x = conc_sec.ultimate_bending_capacity()
ult_res_y = conc_sec.ultimate_bending_capacity(theta=np.pi / 2)
ult_res_x.print_results()
ult_res_y.print_results()
Ultimate Bending Results ┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━┓ ┃ Property ┃ Value ┃ ┡━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━┩ │ Bending Angle - theta │ 0.000000e+00 │ │ Neutral Axis Depth - d_n │ 1.362386e+02 │ │ Neutral Axis Parameter - k_u │ 3.041040e-01 │ │ Axial Force │ -1.865555e+01 │ │ Bending Capacity - m_x │ 9.488647e+08 │ │ Bending Capacity - m_y │ 4.363745e+05 │ │ Bending Capacity - m_xy │ 9.488648e+08 │ └──────────────────────────────┴───────────────┘
Ultimate Bending Results ┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━┓ ┃ Property ┃ Value ┃ ┡━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━┩ │ Bending Angle - theta │ 1.570796e+00 │ │ Neutral Axis Depth - d_n │ 1.714567e+02 │ │ Neutral Axis Parameter - k_u │ 3.827159e-01 │ │ Axial Force │ -2.996934e+01 │ │ Bending Capacity - m_x │ 5.681408e+05 │ │ Bending Capacity - m_y │ -8.450024e+08 │ │ Bending Capacity - m_xy │ 8.450026e+08 │ └──────────────────────────────┴───────────────┘
[10]:
mi_res = conc_sec.moment_interaction_diagram()
mi_res.plot_diagram()
[10]:
<AxesSubplot:title={'center':'Moment Interaction Diagram'}, xlabel='Bending Moment', ylabel='Axial Force'>
[11]:
bb_res = conc_sec.biaxial_bending_diagram()
bb_res.plot_diagram()
[11]:
<AxesSubplot:title={'center':'Biaxial Bending Diagram, $N = 0.000e+00$'}, xlabel='Bending Moment $M_x$', ylabel='Bending Moment $M_y$'>
[12]:
ult_stress_x = conc_sec.calculate_ultimate_stress(ult_res_x)
ult_stress_y = conc_sec.calculate_ultimate_stress(ult_res_y)
ult_stress_x.plot_stress()
ult_stress_y.plot_stress()
[12]:
<AxesSubplot:title={'center':'Stress'}>
8.3. Concrete Filled Steel Column#
Next we will analyse a 323.9 mm x 12.7 mm steel (350 MPa) circular hollow section filled with concrete and 6N20 bars. The results will be compared to a similarly sized concrete column.
[13]:
steel_350 = Steel(
name="350 MPa Structural Steel",
density=7.85e-6,
stress_strain_profile=ssp.SteelElasticPlastic(
yield_strength=350,
elastic_modulus=200e3,
fracture_strain=0.05,
),
colour="tan",
)
[14]:
# create outer diameter of steel column
steel_col = sp_ps.circular_section_by_area(
area=np.pi * 323.9**2 / 4,
n=64,
material=steel_350,
)
# create inner diameter of steel column, concrete filled
inner_conc = sp_ps.circular_section_by_area(
area=np.pi * (323.9 - 2 * 12.7) ** 2 / 4,
n=64,
material=concrete,
)
# create composite geometry
geom_comp = steel_col - inner_conc + inner_conc
# add reinforcement
r_bars = 323.9 / 2 - 12.7 - 30 - 10 # 30 mm cover from inside of steel
geom_comp = cp_pre.add_bar_circular_array(
geometry=geom_comp,
area=310,
material=steel_bar,
n_bar=6,
r_array=r_bars,
)
# create concrete section and plot
conc_sec_comp = ConcreteSection(geom_comp)
conc_sec_comp.plot_section()
# create 350 diameter column for comparison
geom_conc = sp_cs.concrete_circular_section(
d=350,
n=64,
dia=20,
n_bar=6,
n_circle=4,
cover=30 + 12, # 30 mm cover + 12 mm tie
area_conc=np.pi * 350**2 / 4,
area_bar=310,
conc_mat=concrete,
steel_mat=steel_bar,
)
# create concrete section and plot
conc_sec_conc = ConcreteSection(geom_conc)
conc_sec_conc.plot_section()
[14]:
<AxesSubplot:title={'center':'Reinforced Concrete Section'}>
8.3.1. Compare Elastic Stresses#
Compare the elastic stresses under a bending moment of 10 kN.m (uncracked).
[15]:
el_stress_comp = conc_sec_comp.calculate_uncracked_stress(m_x=10e6)
el_stress_conc = conc_sec_conc.calculate_uncracked_stress(m_x=10e6)
el_stress_comp.plot_stress()
el_stress_conc.plot_stress()
[15]:
<AxesSubplot:title={'center':'Stress'}>
Compare the elastic stresses under a bending moment of 50 kN.m (cracked).
[16]:
cr_res_comp = conc_sec_comp.calculate_cracked_properties()
cr_res_conc = conc_sec_conc.calculate_cracked_properties()
cr_stress_comp = conc_sec_comp.calculate_cracked_stress(
cracked_results=cr_res_comp, m=50e6
)
cr_stress_conc = conc_sec_conc.calculate_cracked_stress(
cracked_results=cr_res_conc, m=50e6
)
cr_stress_comp.plot_stress()
cr_stress_conc.plot_stress()
[16]:
<AxesSubplot:title={'center':'Stress'}>
8.3.2. Compare Moment Curvature Response#
Calculate the moment curvature response of each column and compare the results.
[17]:
mk_res_comp = conc_sec_comp.moment_curvature_analysis()
mk_res_conc = conc_sec_conc.moment_curvature_analysis()
[18]:
res.MomentCurvatureResults.plot_multiple_results(
moment_curvature_results=[mk_res_comp, mk_res_conc],
labels=["Composite", "Concrete"],
fmt="-",
)
[18]:
<AxesSubplot:title={'center':'Moment-Curvature'}, xlabel='Curvature', ylabel='Moment'>
Although the composite section is significantly stiffer and stronger, the regular concrete section exhibits a more ductile response due to the reduced amount of steel reinforcement.
8.3.3. Compare Moment Interaction Results#
Moment interaction diagrams can be produced for each section and compared. Extra points are generated on the tensile side of the composite curve to improve the fidelity of the curve.
[19]:
mi_res_comp = conc_sec_comp.moment_interaction_diagram(
control_points=[
("kappa0", 0.0),
("D", 1.0),
("N", 0),
("N", -2e6),
("N", -3e6),
("d_n", 1e-06),
],
n_points=[6, 16, 4, 4, 4],
)
mi_res_conc = conc_sec_conc.moment_interaction_diagram()
res.MomentInteractionResults.plot_multiple_diagrams(
moment_interaction_results=[mi_res_comp, mi_res_conc],
labels=["Composite", "Concrete"],
fmt="x-",
)
[19]:
<AxesSubplot:title={'center':'Moment Interaction Diagram'}, xlabel='Bending Moment', ylabel='Axial Force'>
8.4. Composite Steel Beam#
Finally we will analyse a steel composite beam, consisting of a 530UB92 universal beam compositely attached to a 1500 mm wide x 120 mm deep concrete slab. We start by defining the concrete material (32 MPa) for the slab.
[20]:
concrete = Concrete(
name="32 MPa Concrete",
density=2.4e-6,
stress_strain_profile=ssp.ConcreteLinearNoTension(
elastic_modulus=30.1e3,
ultimate_strain=0.003,
compressive_strength=0.9 * 32,
),
ultimate_stress_strain_profile=ssp.RectangularStressBlock(
compressive_strength=32,
alpha=0.802,
gamma=0.89,
ultimate_strain=0.003,
),
flexural_tensile_strength=3.4,
colour="lightgrey",
)
Next we construct the composite geometry.
[21]:
# create 530UB92
ub = sp_ss.i_section(
d=533,
b=209,
t_f=15.6,
t_w=10.2,
r=14,
n_r=8,
material=steel_300,
)
# create concrete slab, centre on top of UB
conc_slab = (
sp_ps.rectangular_section(
d=120,
b=1500,
material=concrete,
)
.align_center(align_to=ub)
.align_to(other=ub, on="top")
)
# as there is no overlapping geometry, we can simply add the two sections together
geom = ub + conc_slab
# the cross-section is rotated 90 degrees to aid viewing the stress plots
geom = geom.rotate_section(angle=np.pi / 2, use_radians=True)
# create concrete section and plot
conc_sec = ConcreteSection(geom)
conc_sec.plot_section()
[21]:
<AxesSubplot:title={'center':'Reinforced Concrete Section'}>
8.4.1. Moment Curvature Response#
Calculate the moment curvature response of the composite section. The section should show a gradual yielding response as plasticity develops over the depth of the steel beam.
[22]:
mk_res = conc_sec.moment_curvature_analysis(theta=np.pi / 2)
[23]:
mk_res.plot_results()
[23]:
<AxesSubplot:title={'center':'Moment-Curvature'}, xlabel='Curvature', ylabel='Moment'>
We can examine the stresses at various point within the moment-curvature response.
[24]:
el_stress = conc_sec.calculate_service_stress(moment_curvature_results=mk_res, m=500e6)
yield_stress = conc_sec.calculate_service_stress(
moment_curvature_results=mk_res, m=1000e6
)
ult_stress = conc_sec.calculate_service_stress(
moment_curvature_results=mk_res, m=1200e6
)
[25]:
el_stress.plot_stress()
yield_stress.plot_stress()
ult_stress.plot_stress()
[25]:
<AxesSubplot:title={'center':'Stress'}>
8.4.2. Ultimate Bending Capacity#
The ultimate bending capacity can be calculated and compared to the result obtained from the moment curvature analysis.
[26]:
ult_res = conc_sec.ultimate_bending_capacity(theta=np.pi / 2)
ult_res.print_results()
Ultimate Bending Results ┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━┓ ┃ Property ┃ Value ┃ ┡━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━┩ │ Bending Angle - theta │ 1.570796e+00 │ │ Neutral Axis Depth - d_n │ 9.472512e+01 │ │ Neutral Axis Parameter - k_u │ 0.000000e+00 │ │ Axial Force │ 3.399405e+01 │ │ Bending Capacity - m_x │ 3.423445e+05 │ │ Bending Capacity - m_y │ -1.194843e+09 │ │ Bending Capacity - m_xy │ 1.194843e+09 │ └──────────────────────────────┴───────────────┘
The above results match closely with the moment curvature results, as will as confirming that the neutral axis is within the concrete slab. Note that k_u is not calculated for only meshed sections, it is only computed for lumped reinforcement.