def test_deflection_for_fixed_cantilevered_beam_with_load_at_any_point():
P = -1000 # load in lbs down
a = 7 # position of load in inches
L = 25 # length of beam in inches
E = 29e6 # Young's modulus in psi
Ixx = 345 # area moment of inertia in in**4
b = L - a
# function to calculate deflection anywhere along the length of the beam
d1 = lambda x: P * b**2 / (6 * E * Ixx) * (3 * L - 3 * x - b) # x < a
d2 = lambda x: P * (L - x)**2 / (6 * E * Ixx) * (3 * b - L + x) # x > a
d3 = lambda x: P * b**3 / (3 * E * Ixx) # x == a
beam = Beam(L, [PointLoad(P, a)], [FixedReaction(L)], E, Ixx)
for x in [0, 7, 12.5, 25]:
# check the deflection of the beam at both end points, and the center
if x < a:
exact = d1(x)
elif x > a:
exact = d2(x)
else:
exact = d3(x)
assert pytest.approx(beam.deflection(x), rel=1e-12) == exact, \
f"Calculated deflection does not match expected deflection at {x}"
def test_reaction_types():
pr = PinnedReaction(0)
fr = FixedReaction(0)
assert pr.boundary == (
0, None), "PinnedReaction only has one degree of freedom"
assert fr.boundary == (
0, 0), "FixedReaction does not have any degrees of freedom"
def test_stiffness_matrix_k():
beam = Beam(25, [PointLoad(-100, 25)], [FixedReaction(0)], 29e6, 345)
assert beam.K.shape == (4, 4), "stiffness matrix is not expected size"
# add another point load and verify the stiffness matrix changes size
# accordingly
beam.loads.append(PointLoad(-500, 12))
beam.remesh()
assert beam.K.shape == (6, 6), "stiffness matrix size did not update"
def test_node_deflections_at_free_end():
for load in [PointLoad(-100, 25), MomentLoad(-100, 25)]:
beam = Beam(25, [load], [FixedReaction(0)], 29e6, 345)
# check that the deflection at the free end is non-zero and negative
msgs = [
"displacement at free end is not negative",
"angular displacement at free end is not negative",
]
for i, msg in enumerate(msgs, 2):
assert beam.node_deflections[i][0] < 0, msg
def test_node_deflections_at_fixed_end():
for load in [PointLoad(-100, 25), MomentLoad(-100, 25)]:
beam = Beam(25, [load], [FixedReaction(0)], 29e6, 345)
# check that the deflection at the fixed end is 0
msgs = [
"displacement at fixed end is non-zero",
"angular displacement at fixed end is non-zero",
]
for i, msg in enumerate(msgs):
assert beam.node_deflections[i][0] == 0, msg
def test_plot_default_labels():
b = Beam(10, [PointLoad(10, 10)], [FixedReaction(0)])
fig, axes = b.plot()
x_labels = ("", "", "Beam position, x")
y_labels = ("shear", "moment", "deflection")
assert len(axes) == len(x_labels), "wrong number of sub-plots"
for ax, x_label, y_label in zip(axes, x_labels, y_labels):
assert ax.get_xlabel() == x_label
assert ax.get_ylabel() == y_label
def test_invalid_deflection_location():
beam = Beam(25, [PointLoad(-100, 25)], [FixedReaction(0)], 29e6, 345)
with pytest.raises(ValueError):
beam.deflection(-4) # a value less than 0
with pytest.raises(ValueError):
beam.deflection(beam.length + 5) # a value greater then the length
with pytest.raises(TypeError):
beam.deflection('a string (not a number)')
def test_plot_custom_labels():
b = Beam(10, [PointLoad(10, 10)], [FixedReaction(0)])
diagrams = ("deflection", "deflection", "moment", "shear")
labels = ("def1", "def2", "M", "V")
fig, axes = b.plot(diagrams=diagrams, diagram_labels=labels)
assert len(axes) == len(diagrams), "wrong number of sub-plots"
x_labels = ["" for _ in range(len(diagrams) - 1)]
x_labels.append("Beam position, x")
for ax, x_label, y_label in zip(axes, x_labels, labels):
assert ax.get_xlabel() == x_label
assert ax.get_ylabel() == y_label
def test_solve_method():
beam = Beam(25, [PointLoad(-100, 25)], [FixedReaction(0)], 29e6, 345)
reaction = beam.reactions[0]
assert reaction.force is None, "Reaction force was not None before being solved"
assert reaction.moment is None, "Reaction moment was not None before being solved"
beam.solve()
reaction = beam.reactions[0]
assert reaction.force == 100, "Reaction force must be equal to and opposite load"
assert (
reaction.moment == 100 *
25), "Reaction moment must be equal to the load times the moment arm"
def test_apply_boundary_conditions():
beam = Beam(25,
[PointLoad(-100, 25), PointLoad(-100, 12)], [FixedReaction(0)],
29e6, 345)
k = beam.K
bcs = [(None, None), (0, 0)]
initial_shape = beam.K.shape
assert initial_shape == (
6, 6), "stiffness matrix does not match expected size"
ki = beam.apply_boundary_conditions(k, bcs)
final_shape = ki.shape
assert initial_shape == final_shape, ("stiffness matrix changed shape "
"when applying boundary conditions")
def test_shear():
beam = Beam(25, [PointLoad(-1000, 25)], [FixedReaction(0)])
for x in [0.5, 5, 13, 20, 24.5]:
assert pytest.approx(beam.shear(x), rel=1e-5) == 1000, \
f"shear does not equal load at location {x}"
# right now, the derivative function will try to calculate shear outside
# of the beam when calculating shear at or near endpoints. Verify that
# calculating shear at ends raises a ValueError. It should also raise a
# ValueError when the input is outside the beam
for x in [-5, 0, 25, 35]:
with pytest.raises(ValueError):
beam.shear(x)
def test_moment_for_fixed_cantilevered_beam_with_load_at_end():
P = -1000 # load in lbs down
L = 25 # length of beam in inches
E = 29e6 # Young's modulus in psi
Ixx = 345 # area moment of inertia in in**4
# function to calculate moment anywhere along the length of the beam
m = lambda x: P * x
beam = Beam(L, [PointLoad(P, 0)], [FixedReaction(L)], E, Ixx)
for x in [7, 12.5, 25]:
# check the deflection of the beam at both end points, and the center
assert pytest.approx(beam.moment(x), rel=0.01) == m(x), \
f"Calculated moment does not match expected moment at {x}"
with pytest.warns(UserWarning):
beam.moment(0)
def example_1():
"""
Cantilevered Beam with Fixed Support and End Loading
"""
print("=" * 79)
print("Example 1")
print(
"Show an example with a cantilevered beam with a fixed support and "
"point load at the end\n"
)
beam_len = 10
# Note that both the reaction and load are both lists. They must always be
# given to Beam as a list,
r = [FixedReaction(0)] # define reactions as list
p = [PointLoad(magnitude=-2, location=beam_len)] # define loads as list
b = Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)
# an explicit solve is required to calculate the reaction values
b.solve()
print(b)
def test_cantilevered_beam_load_at_end():
"""fixed beam with concentrated load at free end
case 13
"""
R = -P
M_max = P * L # at fixed end
d_max = P * L**3 / (3 * EI) # at free end
beam = Beam(
length=L,
loads=[PointLoad(magnitude=P, location=0)],
reactions=[FixedReaction(L)],
E=E,
Ixx=Ixx,
)
beam.solve()
validate(beam, loc=0, R=[(R, M_max)], M_loc=0, d_loc=d_max)
assert pytest.approx(beam.moment(L), rel=TOL) == M_max
def test_invalid_load_placement():
reactions = [PinnedReaction(x) for x in [0, 50, 100]]
loads = [PointLoad(-100, x) for x in [0, 50, 100]]
# there should be a warning indicating that the load position was moved
# slightly so it does not line up with the reaction.
with pytest.warns(UserWarning):
beam = Beam(100, loads=loads, reactions=reactions)
for load, reaction in zip(beam.loads, beam.reactions):
assert (load.location != reaction.location
), "moved load is still the same as a reaction"
@pytest.mark.parametrize("invalid_load",
["a string", FixedReaction(0), [], 10])
def test_invalid_load_errors(invalid_load):
# Check for a TypeError for a variety of invalid loads
with pytest.raises(TypeError):
Beam(25, loads=[invalid_load], reactions=[FixedReaction(0)])
@pytest.mark.parametrize("invalid_reaction",
["a string", PointLoad(25, 15), [], 10])
def test_invalid_reaction_errors(invalid_reaction):
# Check for an TypeError for a variety of invalid reactions
with pytest.raises(TypeError):
Beam(25, loads=[PointLoad(-100, 15)], reactions=[invalid_reaction])
def test_shape_function():
def beam_fixed(length):
yield Beam(length=length,
loads=[PointLoad(-100, length)],
reactions=[FixedReaction(0)])
def reaction_fixed():
"""common fixed reaction at 0"""
yield [FixedReaction(0)]
def test_bending_stress_depreciation_warning():
with pytest.warns(DeprecationWarning):
b = Beam(10, [PointLoad(10, 10)], [FixedReaction(0)])
b.bending_stress(x=5, c=1)
def test_invalid_load_errors():
# Check for a TypeError for a variety of invalid loads
for invalid_load in ['a string', FixedReaction(0), [], 10]:
with pytest.raises(TypeError):
Beam(25, loads=[invalid_load], reactions=[FixedReaction(0)])
def test_plot_one_diagram():
b = Beam(10, [PointLoad(10, 10)], [FixedReaction(0)])
fig, axes = b.plot(diagrams=("deflection", ))
assert len(axes) == 1, "expected length of axes was 1"
for ax, y_label in zip(axes, ("deflection", )):
assert ax.get_ylabel() == y_label
import pytest
from femethods.loads import PointLoad
from femethods.mesh import Mesh
from femethods.reactions import FixedReaction
LOADS = [PointLoad(magnitude=-1000, location=15)]
REACTIONS = [FixedReaction(0)]
LENGTH = 25
@pytest.fixture
def mesh():
return Mesh(LENGTH, LOADS, REACTIONS, dof=2)
def test_mesh_properties(mesh):
# there should be a node at the start and end of the beam, as well as a
# node for any loads
assert mesh.nodes == [0, 15, 25], "Mesh nodes do not match expected"
assert mesh.dof == 6, "Wrong number of global degrees-of-freedom"
assert mesh.lengths == [15,
10], "Mesh elements do not have expected lengths"
assert mesh.num_elements == 2, "Mesh element count does not match"
assert mesh.num_elements == len(mesh.lengths)
with pytest.raises(AttributeError):
mesh.nodes = "Mesh nodes property is read-only"
with pytest.raises(AttributeError):
mesh.dof = "Mesh dof should be read-only"
with pytest.raises(AttributeError):
def test_plot_diagram_labels_without_diagrams():
with pytest.raises(ValueError):
b = Beam(10, [PointLoad(10, 10)], [FixedReaction(0)])
b.plot(diagram_labels=("V, lb", "M, in/lb", "delta, in"))
def test_plot_diagrams_diagrams_label_mismatch():
with pytest.raises(ValueError):
b = Beam(10, [PointLoad(10, 10)], [FixedReaction(0)])
b.plot(diagrams=("shear", ), diagram_labels=("shear", "moment"))
def test_plot_diagrams_invalid_value():
with pytest.raises(ValueError):
b = Beam(10, [PointLoad(10, 10)], [FixedReaction(0)])
b.plot(diagrams=("shear", "bad value"))
def test_invalid_load_errors(invalid_load):
# Check for a TypeError for a variety of invalid loads
with pytest.raises(TypeError):
Beam(25, loads=[invalid_load], reactions=[FixedReaction(0)])
def test_shape_of_node_deflections():
beam = Beam(25, [PointLoad(-100, 25)], [FixedReaction(0)], 29e6, 345)
assert beam.node_deflections.shape == (4, 1), \
"nodal deflections shape is not expected"
