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test_pilkey.py
1085 lines (923 loc) · 34.5 KB
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test_pilkey.py
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"""Validation tests from Pilkey.
BibTeX reference:
@book{Pilkey,
author = {Pilkey, Walter D},
booktitle = {Analysis and Design of Elastic Beams},
edition = {First},
isbn = {0471381527},
language = {eng},
publisher = {Wiley},
title = {Analysis and Design of Elastic Beams: Computational Methods},
year = {2002},
}
"""
from __future__ import annotations
from typing import Callable
import numpy as np
import pytest
import pytest_check as check
from shapely import LineString, Polygon, buffer
from sectionproperties.analysis import Section
from sectionproperties.pre import Geometry, Material
from sectionproperties.pre.library import (
circular_hollow_section,
circular_section_by_area,
elliptical_hollow_section,
elliptical_section,
rectangular_section,
)
@pytest.fixture
def figure_1_8() -> Callable:
"""Defines the geometry in Figure 1.8 (page 18).
Returns:
Section object
"""
def _generate_sec(a: float, t: float, ms: float) -> Section:
points = [
[-0.5 * t, -2 * a],
[0.5 * t, -2 * a],
[0.5 * t, -0.5 * t],
[a, -0.5 * t],
[a, 0.5 * t],
[-0.5 * t, 0.5 * t],
]
geom = Geometry(Polygon(points))
geom.create_mesh(mesh_sizes=ms)
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
return sec
return _generate_sec
@pytest.fixture
def figure_6_7() -> Callable:
"""Defines the geometry in Figure 6.7 (page 253).
Returns:
Section object
"""
def _generate_sec(c: float, ms: float, mat: Material | None = None) -> Section:
points = [
[-0.05 * c, -2 * c - 0.05 * c],
[c, -2 * c - 0.05 * c],
[c, -2 * c + 0.05 * c],
[0.05 * c, -2 * c + 0.05 * c],
[0.05 * c, -0.05 * c],
[c, -0.05 * c],
[c, 0.05 * c],
[-0.05 * c, 0.05 * c],
]
if mat:
geom = Geometry(Polygon(points), material=mat)
else:
geom = Geometry(Polygon(points))
geom.create_mesh(mesh_sizes=ms)
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
return sec
return _generate_sec
@pytest.fixture
def figure_6_8() -> Callable:
"""Defines the geometry in Figure 6.8 (page 263).
Returns:
Section object
"""
def _generate_sec(
d: float,
t: float,
mat: Material | None = None,
shift: tuple[float, float] | None = None,
) -> Section:
geom = circular_hollow_section(d=d, t=t, n=128)
ms = geom.calculate_area() / 500.0
geom.points[0] = (d / 2, -1e-3)
geom.points[128] = (d / 2 - t, 1e-3)
if mat:
geom = Geometry(Polygon(geom.points), material=mat)
else:
geom = Geometry(Polygon(geom.points))
geom = geom.rotate_section(angle=-90)
if shift:
geom = geom.shift_section(x_offset=shift[0], y_offset=shift[1])
geom.create_mesh(mesh_sizes=[ms])
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
return sec
return _generate_sec
@pytest.fixture
def figure_6_9() -> Callable:
"""Defines the geometry in Figure 6.9 (page 264).
Returns:
Section object
"""
def _generate_sec(beta: float, mat: Material) -> Section:
geom = rectangular_section(d=1.0, b=2.0, material=mat).rotate_section(
angle=beta
)
geom.create_mesh(mesh_sizes=0.01)
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
return sec
return _generate_sec
@pytest.fixture
def al_cu() -> tuple[Material, Material]:
"""Defines the Aluminium and Copper materials.
Aluminium: E = 10.4e6; nu = 0.3
Copper: E = 18.5e6; nu = 0.3
Returns:
Aluminium and Copper material
"""
al = Material("Al", 10.4e6, 0.3, 1, 1, "r")
cu = Material("Cu", 18.5e6, 0.3, 1, 1, "b")
return al, cu
def test_example_1_1(figure_1_8):
"""Validation test for Example 1.1 (page 18).
Note results calculated based on on thin-wall assumptions.
"""
tol = 1e-3
stress_tol = 5e-2 # note stress inaccuracy due to thin wall assumption in Pilkey
a = 1
t = 0.1
ms = 1e-3
sec = figure_1_8(a=a, t=t, ms=ms)
# check section properties
check.almost_equal(sec.section_props.area, 0.3, rel=tol)
check.almost_equal(sec.section_props.qx, -0.1999, rel=tol)
check.almost_equal(sec.section_props.qy, 0.0499, rel=tol)
check.almost_equal(sec.section_props.cx, 0.1663, rel=tol)
check.almost_equal(sec.section_props.cy, -0.6663, rel=tol)
check.almost_equal(sec.section_props.ixx_c, 0.1336, rel=tol)
check.almost_equal(sec.section_props.iyy_c, 0.0252, rel=tol)
check.almost_equal(sec.section_props.ixy_c, 0.0332, rel=tol)
check.almost_equal(sec.section_props.phi, -15.7589, rel=tol)
check.almost_equal(sec.section_props.i11_c, 0.1430, rel=tol)
check.almost_equal(sec.section_props.i22_c, 0.0158, rel=2 * tol)
# check stresses
pts = [(a, 0), (0, 0), (0, -2 * a)]
sigs = sec.get_stress_at_points(pts=pts, mxx=-1)
check.almost_equal(sigs[0][0], 5, rel=stress_tol)
check.almost_equal(sigs[1][0], -10, rel=stress_tol)
check.almost_equal(sigs[2][0], 12.5, rel=stress_tol)
@pytest.mark.parametrize("d", [1.0, 50.0, 150.0])
def test_example_5_1(d: float):
"""Validation test for Example 5.1 (page 174).
Note higher tolerance due to circle discretisation.
J = Ixx + Iyy
J = pi * d**4 / 32
tau_zx = -Mzz * y / J
tau_zy = Mzz * x / J
CHS -> J = pi / 32 * (do**4 - di**4)
"""
tol = 1e-3
# create solid geometry
area = np.pi * d**2 / 4
ms = area / 500.0
geom_solid = circular_section_by_area(area=area, n=64)
geom_solid.create_mesh(mesh_sizes=[ms])
sec_solid = Section(geometry=geom_solid)
# conduct solid analysis
sec_solid.calculate_geometric_properties()
sec_solid.calculate_warping_properties()
# check J results
j_numerical = sec_solid.get_j()
check.almost_equal(j_numerical, sum(sec_solid.get_ic()[:2]), rel=tol)
check.almost_equal(j_numerical, np.pi * d**4 / 32, rel=tol)
# conduct stress analysis
pts = [(d / 2, 0), (0, d / 2)]
mzz = d * d
sigs = sec_solid.get_stress_at_points(pts=pts, mzz=mzz)
# check stress results
check.almost_equal(sigs[0][1], 0, abs=tol)
check.almost_equal(sigs[0][2], mzz * d / 2 / j_numerical, rel=tol)
check.almost_equal(sigs[1][1], -mzz * d / 2 / j_numerical, rel=tol)
check.almost_equal(sigs[1][2], 0, abs=tol)
# create CHS geometry
geom_chs = circular_hollow_section(d=d, t=0.05 * d, n=64)
geom_chs.create_mesh(mesh_sizes=[ms / 2])
sec_chs = Section(geometry=geom_chs)
# conduct solid analysis
sec_chs.calculate_geometric_properties()
sec_chs.calculate_frame_properties()
# check J results
d_in = d - 0.1 * d
check.almost_equal(sec_chs.get_j(), np.pi / 32 * (d**4 - d_in**4), rel=1e-2)
@pytest.mark.parametrize(("a", "b"), [(50, 75), (75, 50), (10, 50)])
def test_example_5_3(a: float, b: float):
"""Validation test for Example 5.3 (page 176).
Note higher tolerance due to circle discretisation.
J = pi * a**3 * b**3 / (a**2 + b**2)
tau_zx = -2 * Mzz * y / (a * b**3 * pi)
tau_zy = 2 * Mzz * x / (a**3 * b * pi)
"""
tol = 1e-2
# create geometry
geom = elliptical_section(d_x=2 * a, d_y=2 * b, n=64)
ms = geom.calculate_area() / 500.0
geom.create_mesh(mesh_sizes=[ms])
sec = Section(geometry=geom)
# conduct analysis
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
# check J results
j_theory = np.pi * a**3 * b**3 / (a**2 + b**2)
check.almost_equal(sec.get_j(), j_theory, rel=tol)
# conduct stress analysis
pts = [(a, 0), (0, b * 0.75)]
mzz = a * b
sigs = sec.get_stress_at_points(pts=pts, mzz=mzz)
# check stress results
factor_x = -2 / (a * b**3 * np.pi)
factor_y = 2 / (a**3 * b * np.pi)
check.almost_equal(sigs[0][1], 0, abs=tol)
check.almost_equal(sigs[0][2], factor_y * mzz * a, rel=2 * tol)
check.almost_equal(sigs[1][1], factor_x * mzz * b * 0.75, rel=2 * tol)
check.almost_equal(sigs[1][2], 0, abs=tol)
def test_example_5_6():
"""Validation test for Example 5.6 (page 184).
J = h**4 / (15 * sqrt(3))
"""
tol = 1e-6
a = 1 # base length
h = a * np.sqrt(3) / 2
# create geometry
geom = Geometry(Polygon([(-a / 2, 0), (a / 2, 0), (0, h)]))
ms = geom.calculate_area() / 1000.0
geom.create_mesh(mesh_sizes=[ms])
sec = Section(geometry=geom)
# conduct analysis and check j
sec.calculate_frame_properties()
check.almost_equal(sec.get_j(), h**4 / (15 * np.sqrt(3)), rel=tol)
def test_example_5_7(figure_6_8):
"""Validation test for Example 5.7 (page 189).
8 x 0.25 CHS + same with slit. Note higher tolerance due to circle discretisation.
J_chs = 91.49
J_slit = 0.13
tau_chs = 0.0437
tau_slit = 1.92
"""
tol = 5e-2
# create chs
chs = circular_hollow_section(d=8, t=0.25, n=64)
ms = chs.calculate_area() / 500.0
chs.create_mesh(mesh_sizes=[ms])
sec_chs = Section(geometry=chs)
# conduct analysis and check results
sec_chs.calculate_geometric_properties()
sec_chs.calculate_warping_properties()
check.almost_equal(sec_chs.get_j(), 91.49, rel=tol)
sigs = sec_chs.get_stress_at_points(pts=[(0, 4)], mzz=1)
check.almost_equal(sigs[0][1], -0.0437, rel=tol)
# create chs with slit
sec_slit_chs = figure_6_8(d=8, t=0.25)
# conduct analysis and check results
sec_slit_chs.calculate_warping_properties()
check.almost_equal(sec_slit_chs.get_j(), 0.13, rel=tol)
sigs = sec_slit_chs.get_stress_at_points(pts=[(0, 4)], mzz=1)
check.almost_equal(sigs[0][1], -1.92, rel=tol)
def test_example_5_13(al_cu: tuple[Material, Material]):
"""Validation test for Example 5.13 - Composite Rectangular Strip (page 220).
2 off 15 x 2 bars (1 Aluminium, 1 Copper)
EA_al = 83.36
J_al = 106.22
"""
al, cu = al_cu
tol = 1e-3
# create geometry
geom_al = rectangular_section(d=2, b=15, material=al)
geom_cu = rectangular_section(d=2, b=15, material=cu).align_to(geom_al, "right")
geom = geom_al + geom_cu
geom.create_mesh(mesh_sizes=[0.1])
sec = Section(geometry=geom)
# analysis and results
sec.calculate_frame_properties()
check.almost_equal(sec.get_ea(e_ref=al), 83.36, rel=tol)
check.almost_equal(sec.get_ej(e_ref=al), 106.22, rel=tol)
def test_example_5_14(figure_1_8):
"""Validation test for Example 5.14 (page 221).
a = 1, t = 0.1
J = 0.00099
tau = 98e3 from mzz=1e3
"""
tol = 5e-3
stress_tol = 5e-2
# define geometry
sec = figure_1_8(a=1, t=0.1, ms=1e-3)
# analysis and results
sec.calculate_warping_properties()
# check stress at top mid-point along edge
sigs = sec.get_stress_at_points(pts=[(0.5, -0.05)], mzz=1e3)
check.almost_equal(sec.get_j(), 0.00099, rel=tol)
check.almost_equal(sigs[0][1], 98e3, rel=stress_tol)
def test_example_6_1(figure_1_8):
"""Validation test for Example 6.1 (page 233).
a = 10, t = 1
tau_min = P / (12 * a)
tau_max = 25 * P / (36 * a)
"""
tol = 1e-2
# define geometry
sec = figure_1_8(a=10, t=1, ms=0.1)
# analysis and results
sec.calculate_warping_properties()
sigs = sec.calculate_stress(vy=-1).get_stress()[0]
vx = sigs["sig_zx_vy"]
vy = sigs["sig_zy_vy"]
# check stresses
v_min = np.sqrt(max(vx) ** 2 + max(vy) ** 2) # assume occurs at same spot
check.almost_equal(v_min, 1 / (12 * 10), rel=5 * tol)
check.almost_equal(min(vy), -25 * 1 / (36 * 10), rel=tol) # vx insignificant
def test_example_6_2(figure_6_7):
"""Validation test for Example 6.2 (page 253)."""
tol = 1e-3
# define geometry
mat = Material("a", 1, 0.3, 1, 1, "k")
sec = figure_6_7(c=1.0, ms=1e-3, mat=mat)
# analysis and results
sec.calculate_warping_properties()
x_c, y_c = sec.get_c()
x_sc, y_sc = sec.get_sc()
area = sec.get_area()
a_sx, a_sy = sec.get_eas()
check.almost_equal(x_c, 0.24937, rel=tol)
check.almost_equal(y_c, -1.0, rel=tol)
check.almost_equal(x_sc - x_c, -0.62056, rel=tol)
check.almost_equal(y_sc - y_c, 0.0, abs=tol)
check.almost_equal(area / a_sx, 3.09621, rel=tol)
check.almost_equal(area / a_sy, 2.34102, rel=tol)
check.almost_equal(sec.get_ej(), 0.00133, rel=tol)
@pytest.mark.parametrize(
("nu", "result"), [(0.0, -0.62055), (0.333, -0.62056), (0.5, -0.62057)]
)
def test_example_6_3(figure_6_7, nu: float, result: float):
"""Validation test for Example 6.3 (page 256)."""
tol = 1e-3
# define geometry and perform analysis
mat = Material("a", 1, nu, 1, 1, "k")
sec = figure_6_7(c=1.0, ms=1e-3, mat=mat)
sec.calculate_warping_properties()
# check results
x_c, _ = sec.get_c()
x_sc, _ = sec.get_sc()
x_sct, _ = sec.get_sc()
check.almost_equal(x_sc - x_c, result, rel=tol)
check.almost_equal(x_sct - x_c, -0.62055, rel=tol)
@pytest.mark.parametrize(("nu", "nu_idx"), [(0.0, 0), (0.3, 1), (0.5, 2)])
@pytest.mark.parametrize(
("beta", "beta_idx"), [(0.0, 0), (30.0, 1), (45.0, 2), (90.0, 3)]
)
def test_example_6_5(figure_6_9, nu: float, nu_idx: int, beta: float, beta_idx: int):
"""Validation test for Example 6.5 (page 264).
Note that the table entry for beta = 60.0 should read beta = 45.0 (note that a_x =
a_y in the results), error in text.
Note that the a_xy values for beta = 30.0 out by a factor of 10, error in text.
"""
results = [
# nu = 0.0
[
{"a_x": 1.2, "a_y": 1.2, "a_xy": 0.0}, # beta = 0.0
{"a_x": 1.2, "a_y": 1.2, "a_xy": 0.0}, # beta = 30.0
{"a_x": 1.2, "a_y": 1.2, "a_xy": 0.0}, # beta = 45.0
{"a_x": 1.2, "a_y": 1.2, "a_xy": 0.0}, # beta = 90.0
],
# nu = 0.3
[
{"a_x": 1.20056, "a_y": 1.27479, "a_xy": 0.0}, # beta = 0.0
{"a_x": 1.21912, "a_y": 1.25624, "a_xy": -0.032142}, # beta = 30.0
{"a_x": 1.23768, "a_y": 1.23768, "a_xy": -0.03711}, # beta = 45.0
{"a_x": 1.27479, "a_y": 1.20056, "a_xy": 0.0}, # beta = 90.0
],
# nu = 0.5
[
{"a_x": 1.20118, "a_y": 1.35605, "a_xy": 0.0}, # beta = 0.0
{"a_x": 1.23990, "a_y": 1.31733, "a_xy": -0.067062}, # beta = 30.0
{"a_x": 1.27861, "a_y": 1.27861, "a_xy": -0.07744}, # beta = 45.0
{"a_x": 1.35605, "a_y": 1.20118, "a_xy": 0.0}, # beta = 90.0
],
]
# set tol and get result
tol = 1e-3
result = results[nu_idx][beta_idx]
# define materials
mat = Material("a", 1, nu, 1, 1, "w")
# define geometry and perform analysis
sec = figure_6_9(beta=beta, mat=mat)
# check results
area = sec.get_area()
a_sx, a_sy = sec.get_eas()
a_sxy = sec.section_props.a_sxy
assert a_sxy
check.almost_equal(area / a_sx, result["a_x"], rel=tol)
check.almost_equal(area / a_sy, result["a_y"], rel=tol)
# check if a_xy is zero, adjust tolerance
if np.isclose(result["a_xy"], 0):
check.almost_equal(area / a_sxy, result["a_xy"], abs=tol)
else:
check.almost_equal(area / a_sxy, result["a_xy"], rel=tol)
@pytest.mark.parametrize(("nu", "idx"), [(0.0, 0), (0.3, 1), (0.5, 2)])
def test_example_6_6(nu: float, idx: int):
"""Validation test for Example 6.6 (page 267)."""
results = [
# nu = 0.0
{
"a_x": 1.313417,
"a_y": 1.169516,
"a_xy": -0.063785,
"x_sc": -0.069460,
"y_sc": -0.072013,
"phi": -4.860108,
"a_11": 1.323148,
"a_22": 1.159780,
},
# nu = 0.3
{
"a_x": 1.465389,
"a_y": 1.170704,
"a_xy": -0.076064,
"x_sc": -0.071690,
"y_sc": -0.090135,
"phi": -4.860108,
"a_11": 1.476062,
"a_22": 1.159977,
},
# nu = 0.5
{
"a_x": 1.630496,
"a_y": 1.171995,
"a_xy": -0.089406,
"x_sc": -0.072681,
"y_sc": -0.098190,
"phi": -4.860108,
"a_11": 1.642190,
"a_22": 1.160191,
},
]
# set tol and get result
tol = 1e-3
result = results[idx]
# define materials
mat = Material("a", 1, nu, 1, 1, "w")
# define geometry and perform analysis
geom = Geometry(Polygon([(-0.5, -1), (0.5, -1), (0.5, 2), (-0.5, 1)]), material=mat)
geom.create_mesh(mesh_sizes=[0.01])
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
# check results
area = sec.get_area()
c_x, c_y = sec.get_c()
x_sc, y_sc = sec.get_sc()
a_sx, a_sy = sec.get_eas()
a_s11, a_s22 = sec.get_eas_p()
a_sxy = sec.section_props.a_sxy
assert a_sxy
check.almost_equal(area / a_sx, result["a_x"], rel=tol)
check.almost_equal(area / a_sy, result["a_y"], rel=tol)
check.almost_equal(area / a_sxy, result["a_xy"], rel=tol)
check.almost_equal(x_sc - c_x, result["x_sc"], rel=tol)
check.almost_equal(y_sc - c_y, result["y_sc"], rel=tol)
check.almost_equal(sec.get_phi(), result["phi"], rel=tol)
check.almost_equal(area / a_s11, result["a_11"], rel=tol)
check.almost_equal(area / a_s22, result["a_22"], rel=tol)
@pytest.mark.parametrize(("nu", "idx"), [(0.0, 0), (0.3, 1), (0.5, 2)])
@pytest.mark.parametrize("sym", [True, False])
def test_example_6_7(nu: float, idx: int, sym: bool):
"""Validation test for Example 6.7 (page 268)."""
results_sym = [
# nu = 0.0
{
"j": 6.539189,
"a_x": 2.301059,
"a_y": 2.301059,
"a_xy": 0.014969,
"x_sc": -2.464750,
"y_sc": -2.464750,
"phi": -135.0, # converted to degrees
},
# nu = 0.3
{
"j": 6.539189,
"a_x": 2.302935,
"a_y": 2.302935,
"a_xy": 0.016782,
"x_sc": -2.464971,
"y_sc": -2.464971,
"phi": -135.0, # converted to degrees
},
# nu = 0.5
{
"j": 6.539189,
"a_x": 2.304973,
"a_y": 2.304973,
"a_xy": 0.018753,
"x_sc": -2.465070,
"y_sc": -2.465070,
"phi": -135.0, # converted to degrees
},
]
results_unsym = [
# nu = 0.0
{
"j": 8.211705,
"a_x": 3.058207,
"a_y": 1.898375,
"a_xy": 0.039510,
"x_sc": -1.997641,
"y_sc": -4.423925,
"phi": 0.430248 * 180 / np.pi - 180.0, # converted to degrees
},
# nu = 0.3
{
"j": 8.211705,
"a_x": 3.061764,
"a_y": 1.899138,
"a_xy": 0.041123,
"x_sc": -1.997605,
"y_sc": -4.424391,
"phi": 0.430248 * 180 / np.pi - 180.0, # converted to degrees
},
# nu = 0.5
{
"j": 8.211705,
"a_x": 3.065628,
"a_y": 1.899967,
"a_xy": 0.042876,
"x_sc": -1.997589,
"y_sc": -4.424592,
"phi": 0.430248 * 180 / np.pi - 180.0, # converted to degrees
},
]
# set tol and get result
tol = 1e-3
if sym:
result = results_sym[idx]
else:
result = results_unsym[idx]
# define materials
mat = Material("a", 1, nu, 1, 1, "w")
# define geometry and perform analysis
if sym:
up1 = (1, 10.5)
up2 = (0, 10.5)
else:
up1 = (1, 15.5)
up2 = (0, 15.5)
geom = Geometry(
Polygon([(0, 0), (10.5, 0), (10.5, 1), (1, 1), up1, up2]), material=mat
)
geom.create_mesh(mesh_sizes=[0.1])
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
# check results
area = sec.get_area()
c_x, c_y = sec.get_c()
x_sc, y_sc = sec.get_sc()
a_sx, a_sy = sec.get_eas()
a_sxy = sec.section_props.a_sxy
assert a_sxy
check.almost_equal(sec.get_ej(), result["j"], rel=tol)
check.almost_equal(area / a_sx, result["a_x"], rel=tol)
check.almost_equal(area / a_sy, result["a_y"], rel=tol)
check.almost_equal(area / a_sxy, result["a_xy"], rel=0.02) # within 2% error
check.almost_equal(x_sc - c_x, result["x_sc"], rel=tol)
check.almost_equal(y_sc - c_y, result["y_sc"], rel=tol)
check.almost_equal(sec.get_phi(), result["phi"], rel=tol)
def test_example_b_1():
"""Validation test for Example B.1 - Closed Elliptical Tube (page 434)."""
# define input values
a_mid = 8 # x-distance from centre to mid-line of EHS
b_mid = 5 # y-distance from centre to mid-line of EHS
t = 1 # thickness of EHS
mat = Material("a", 2.1e8, 0.33333, 1, 1, "w")
tol = 2e-2 # note differences in spline approach used by Pilkey (not an ellipse)
# create geometry, mesh, section, and run analysis
geom = elliptical_hollow_section(
d_x=2 * a_mid + t,
d_y=2 * b_mid + t,
t=t,
n=128,
material=mat,
)
geom.create_mesh(mesh_sizes=[0.1])
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
# conduct checks
ixx, iyy, ixy = sec.get_eic(e_ref=mat)
zxx, _, zyy, _ = sec.get_ez(e_ref=mat)
rx, ry = sec.get_rc()
a_sx, a_sy = sec.get_eas(e_ref=mat)
a_sxy = sec.section_props.a_sxy
assert a_sxy
check.almost_equal(sec.get_area(), 41.38626, rel=tol)
check.almost_equal(ixx, 580.42697, rel=tol)
check.almost_equal(iyy, 1180.33120, rel=tol)
check.almost_equal(ixy, 0, abs=1e-3)
check.almost_equal(zxx, 105.54076, rel=tol)
check.almost_equal(zyy, 138.86247, rel=tol)
check.almost_equal(rx, 3.74495, rel=tol)
check.almost_equal(ry, 5.34040, rel=tol)
check.almost_equal(sec.get_phi(), -90.0, rel=tol)
check.almost_equal(sec.get_area() / a_sx, 1.51457, rel=tol)
check.almost_equal(sec.get_area() / a_sy, 3.05985, rel=tol)
check.almost_equal(sec.get_ea() / a_sxy, 0.0, abs=1e-3)
check.almost_equal(sec.get_ej(e_ref=mat), 1537.38165, rel=tol)
# gamma is within 7%, possible spline error amplifies this?
check.almost_equal(sec.get_egamma(e_ref=mat), 451.90976, rel=7e-2)
def test_example_b_2():
"""Validation test for Example B.2 - Symmetric Channel Section (page 437)."""
# define input values
b = 8
h = 18
t = 1
mat = Material("a", 2.1e8, 0.33333, 1, 1, "w")
tol = 1e-5
warp_tol = 1e-3
zero_tol = 1e-3
# create geometry, mesh, section, and run analysis
geom = Geometry(
Polygon(
[
(-0.5 * t, -0.5 * h - 0.5 * t),
(b, -0.5 * h - 0.5 * t),
(b, -0.5 * h + 0.5 * t),
(0.5 * t, -0.5 * h + 0.5 * t),
(0.5 * t, 0.5 * h - 0.5 * t),
(b, 0.5 * h - 0.5 * t),
(b, 0.5 * h + 0.5 * t),
(-0.5 * t, 0.5 * h + 0.5 * t),
]
),
material=mat,
)
geom.create_mesh(mesh_sizes=[0.1])
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
# conduct checks
qx, qy = sec.get_eq(e_ref=mat)
cx, cy = sec.get_c()
ixx_g, iyy_g, ixy_g = sec.get_eig(e_ref=mat)
ixx_c, iyy_c, ixy_c = sec.get_eic(e_ref=mat)
zxx_p, zxx_m, zyy_p, zyy_m = sec.get_ez(e_ref=mat)
rx, ry = sec.get_rc()
x_sc, y_sc = sec.get_sc()
x_sct, y_sct = sec.get_sc_t()
a_sx, a_sy = sec.get_eas(e_ref=mat)
a_sxy = sec.section_props.a_sxy
assert a_sxy
check.almost_equal(sec.get_area(), 34.0, rel=tol)
check.almost_equal(qx, 0.0, abs=zero_tol)
check.almost_equal(qy, 63.75, rel=tol)
check.almost_equal(cx, 1.875, rel=tol)
check.almost_equal(cy, 0, abs=zero_tol)
check.almost_equal(x_sc, -2.86769, rel=warp_tol)
check.almost_equal(y_sc, 0, abs=zero_tol)
check.almost_equal(x_sct, -2.86759, rel=warp_tol)
check.almost_equal(y_sct, 0, abs=zero_tol)
# textbook erroneously printed 1 instead of 7, ixx_g should equal ixx_c
check.almost_equal(ixx_g, 1787.83333, rel=tol)
check.almost_equal(iyy_g, 342.83333, rel=tol)
check.almost_equal(ixy_g, 0.0, abs=zero_tol)
check.almost_equal(ixx_c, 1787.83333, rel=tol)
check.almost_equal(iyy_c, 223.30208, rel=tol)
check.almost_equal(ixy_c, 0.0, abs=zero_tol)
check.almost_equal(min(zxx_p, zxx_m), 188.19298, rel=tol)
check.almost_equal(min(zyy_p, zyy_m), 36.45748, rel=tol)
check.almost_equal(rx, 7.25144, rel=tol)
check.almost_equal(ry, 2.56275, rel=tol)
check.almost_equal(sec.get_phi(), 0.0, abs=zero_tol)
check.almost_equal(sec.get_area() / a_sx, 3.40789, rel=warp_tol)
check.almost_equal(sec.get_area() / a_sy, 2.15337, rel=warp_tol)
check.almost_equal(sec.get_ea() / a_sxy, 0.0, abs=zero_tol)
check.almost_equal(sec.get_ej(e_ref=mat), 11.28862, rel=warp_tol)
check.almost_equal(sec.get_egamma(e_ref=mat), 12763.15184, rel=warp_tol)
def test_example_b_3():
"""Validation test for Example B.3 - L Section Without Symmetry (page 441)."""
# define input values
b = 5.625
h = 7.625
t = 0.75
mat = Material("a", 2.1e8, 0.33333, 1, 1, "w")
tol = 1e-5
warp_tol = 1e-3
# create geometry, mesh, section, and run analysis
geom = Geometry(
Polygon(
[
(-0.5 * t, -0.5 * t),
(b, -0.5 * t),
(b, 0.5 * t),
(0.5 * t, 0.5 * t),
(0.5 * t, h),
(-0.5 * t, h),
]
),
material=mat,
)
geom.create_mesh(mesh_sizes=[0.05])
sec = Section(geometry=geom)
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
# conduct checks
qx, qy = sec.get_eq(e_ref=mat)
cx, cy = sec.get_c()
ixx_c, iyy_c, ixy_c = sec.get_eic(e_ref=mat)
x_sc, y_sc = sec.get_sc()
i11_c, i22_c = sec.get_eip(e_ref=mat)
check.almost_equal(sec.get_area(), 9.93750, rel=tol)
check.almost_equal(qx, 21.75000, rel=tol)
check.almost_equal(qy, 11.81250, rel=tol)
check.almost_equal(cx, 1.18868, rel=tol)
check.almost_equal(cy, 2.18868, rel=tol)
check.almost_equal(x_sc, 0.00313, rel=10 * warp_tol) # lack of precision in digits
check.almost_equal(y_sc, 0.05735, rel=warp_tol)
check.almost_equal(ixx_c, 63.42455, rel=tol)
check.almost_equal(iyy_c, 30.72142, rel=tol)
check.almost_equal(ixy_c, -25.85377, rel=tol)
check.almost_equal(sec.get_phi(), 28.84407 - 180.0, rel=tol)
check.almost_equal(i11_c, 77.66369, rel=tol)
check.almost_equal(i22_c, 16.48228, rel=tol)
def test_example_b_4(figure_6_8):
"""Validation test for Example B.4 - Open Circular Cross Section (page 444)."""
# define input values
r_mid = 8
t = 1.25
mat = Material("a", 2.1e8, 0.33333, 1, 1, "w")
tol = 1e-3
warp_tol = 1e-3
zero_tol = 1e-2
# define geometry and perform analysis
sec = figure_6_8(d=2 * r_mid + t, t=t, mat=mat, shift=(0.0, 8.0))
sec.calculate_warping_properties()
# conduct checks
qx, qy = sec.get_eq(e_ref=mat)
cx, cy = sec.get_c()
ixx_g, iyy_g, ixy_g = sec.get_eig(e_ref=mat)
ixx_c, iyy_c, ixy_c = sec.get_eic(e_ref=mat)
zxx_p, zxx_m, zyy_p, zyy_m = sec.get_ez(e_ref=mat)
rx, ry = sec.get_rc()
x_sc, y_sc = sec.get_sc()
x_sct, y_sct = sec.get_sc_t()
a_sx, a_sy = sec.get_eas(e_ref=mat)
a_sxy = sec.section_props.a_sxy
assert a_sxy
check.almost_equal(sec.get_area(), 62.83182, rel=tol)
check.almost_equal(qx, 502.65457, rel=tol)
check.almost_equal(qy, 0.0, abs=zero_tol)
check.almost_equal(cx, 0.0, abs=zero_tol)
check.almost_equal(cy, 8.0, rel=tol)
check.almost_equal(x_sc, 0.0, abs=zero_tol)
check.almost_equal(y_sc, 23.90306, rel=warp_tol)
check.almost_equal(x_sct, 0.0, abs=zero_tol)
check.almost_equal(y_sct, 23.90282, rel=warp_tol)
check.almost_equal(ixx_g, 6044.12555, rel=tol)
check.almost_equal(iyy_g, 2022.88907, rel=tol)
check.almost_equal(ixy_g, 0.0, abs=zero_tol)
check.almost_equal(ixx_c, 2022.88890, rel=tol)
check.almost_equal(iyy_c, 2022.88907, rel=tol)
check.almost_equal(ixy_c, 0.0, abs=0.1)
check.almost_equal(min(zxx_p, zxx_m), 234.53784, rel=tol)
check.almost_equal(min(zyy_p, zyy_m), 234.53789, rel=tol)
check.almost_equal(rx, 5.67409, rel=tol)
check.almost_equal(ry, 5.67409, rel=tol)
check.almost_equal(sec.get_area() / a_sx, 5.93977, rel=warp_tol)
check.almost_equal(sec.get_area() / a_sy, 1.98015, rel=warp_tol)
check.almost_equal(sec.get_ea() / a_sxy, 0.0, abs=zero_tol)
check.almost_equal(sec.get_ej(e_ref=mat), 32.23967, rel=2 * warp_tol)
check.almost_equal(sec.get_egamma(e_ref=mat), 331651.29223, rel=2 * warp_tol)
def test_example_b_5():
"""Validation test for Example B.5 - Welded Hat Section (page 445)."""
# define input values
mat = Material("a", 2.1e8, 0.33333, 1, 1, "w")
tol = 5e-2 # note differences in modelling and mesh (pilkey uses splines)
zero_tol = 1e-3
# define geometry and perform analysis
bot_plate = rectangular_section(d=0.75, b=31, material=mat).shift_section(
y_offset=-0.75
)
top_plate_l = rectangular_section(d=0.75, b=6, material=mat)
top_plate_r = rectangular_section(d=0.75, b=6, material=mat).shift_section(
x_offset=25
)
sep = 1e-3 # separation between hat and bottom plate to simulate lack of weld
# hat centreline
line_pts = [
(6, 0.375),
(8, 0.375 + sep),
(12, 6),
(19, 6),
(23, 0.375 + sep),
(25, 0.375),
]
line = LineString(coordinates=line_pts)
# buffer hat centreline
poly = buffer(geometry=line, distance=0.375, cap_style="flat", join_style="mitre")
hat = Geometry(geom=poly, material=mat)
# we must correct the ends of the hat to make sure they are orthogonal to x-y axes
hat.points[4] = (25, 0.75)
hat.points[5] = (25, 0)
hat.points[10] = (6, 0)
hat.points[11] = (6, 0.75)
geom = bot_plate + hat + top_plate_l + top_plate_r
geom.holes = [geom.holes[1]] # remove erroneous holes
geom.create_mesh(mesh_sizes=[0.1])
sec = Section(geometry=geom)
# perform analysis
sec.calculate_geometric_properties()
sec.calculate_warping_properties()
# conduct checks
qx, qy = sec.get_eq(e_ref=mat)
cx, cy = sec.get_c()
ixx_g, iyy_g, ixy_g = sec.get_eig(e_ref=mat)
ixx_c, iyy_c, ixy_c = sec.get_eic(e_ref=mat)
zxx_p, zxx_m, zyy_p, zyy_m = sec.get_ez(e_ref=mat)
rx, ry = sec.get_rc()
x_sc, y_sc = sec.get_sc()
a_sx, a_sy = sec.get_eas(e_ref=mat)
a_sxy = sec.section_props.a_sxy
assert a_sxy
check.almost_equal(sec.get_area(), 50.10830, rel=tol)
check.almost_equal(qx, 57.90772, rel=tol)
check.almost_equal(qy, 776.70074, rel=tol)
check.almost_equal(cx, 15.50044, rel=tol)
check.almost_equal(cy, 1.15565, rel=tol)
check.almost_equal(x_sc, 15.50097, rel=tol)