class UniformCantileverWithLinearTwist_Test:
"""Values from R.H. MacNeal, R.L. Harder: "Proposed standard set of
problems to test FE accuracy"
"""
length = 12.0
width = 1.1
depth = 0.32
E = 29.0e6
tip_twist = np.pi / 2
num_elements = 12
def setup(self):
x = np.linspace(0, self.length, self.num_elements)
twist = x * self.tip_twist / self.length
EIy = self.E * self.width * self.depth**3 / 12
EIz = self.E * self.depth * self.width**3 / 12
self.fe = BeamFE(x, 0, 0, EIy, EIz, twist=twist)
self.fe.set_boundary_conditions('C', 'F')
def test_deflection_under_tip_load_y(self):
Q = transverse_load(self.num_elements, magnitude=1, only_tip=True)
defl, reactions = self.fe.static_deflection(Q=Q)
assert_allclose(defl[-6:][1], 0.001754, atol=1e-5)
def test_deflection_under_tip_load_z(self):
Q = transverse_load(self.num_elements,
magnitude=1,
angle=(np.pi / 2),
only_tip=True)
defl, reactions = self.fe.static_deflection(Q=Q)
assert_allclose(defl[-6:][2], 0.005424, atol=1e-5)
def setUp(self):
# FE model for beam - no modes, i.e. rigid
x = linspace(0, self.length, 20)
density = (2 * self.mass / self.length) * (1 - x / self.length)
fe = BeamFE(x, density=density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
self.beam = ModalElementFromFE('beam', fe, 0)
# Set loading - in Z direction
load = np.zeros((len(x), 3))
load[:, 2] = self.force
self.beam.loading = load
# Offset from hinge axis
self.conn = RigidConnection('offset', [self.offset, 0, 0])
# Hinge with axis along Y axis
self.hinge = Hinge('hinge', [0, 1, 0])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.conn)
self.conn.add_leaf(self.beam)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
def _mock_rigid_uniform_beam(density, length, name='beam'):
"""Return empty modal representation of 20m long rigid beam"""
x = linspace(0, length, 20)
fe = BeamFE(x, density=density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
beam = ModalElementFromFE(name, fe, 0)
return beam
def setUp(self):
# FE model for beam
x = linspace(0, self.length, 20)
fe = BeamFE(x, density=self.density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
self.beam = ModalElementFromFE('beam', fe, 0)
# Set loading - in negative Z direction
load = np.zeros((len(x), 3))
load[:, 2] = -self.force
self.beam.loading = load
# Hinge with axis along Y axis
self.hinge = Hinge('hinge', [0, 1, 0])
self.hinge.internal_torque = self.hinge_torque
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.beam)
self.system.setup()
if not self.free_beam:
# Prescribe hinge to be fixed
self.system.prescribe(self.hinge)
# Initial calculations
self.recalc()
def setUp(self):
# FE model for beam - no modes, i.e. rigid
x = linspace(0, self.length, 20)
density = (2 * self.mass / self.length) * (1 - x / self.length)
fe = BeamFE(x, density=density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
self.beam = ModalElementFromFE('beam', fe, 0)
# Set loading - in Z direction
load = np.zeros((len(x), 3))
load[:, 2] = self.force
self.beam.loading = load
# Offset from hinge axis
self.conn = RigidConnection('offset', [self.offset, 0, 0])
# Hinge with axis along Y axis
self.hinge = Hinge('hinge', [0, 1, 0])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.conn)
self.conn.add_leaf(self.beam)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
class UniformCantileverWithLinearTwist_Test:
"""Values from R.H. MacNeal, R.L. Harder: "Proposed standard set of
problems to test FE accuracy"
"""
length = 12.0
width = 1.1
depth = 0.32
E = 29.0e6
tip_twist = np.pi / 2
num_elements = 12
def setup(self):
x = np.linspace(0, self.length, self.num_elements)
twist = x * self.tip_twist / self.length
EIy = self.E * self.width * self.depth ** 3 / 12
EIz = self.E * self.depth * self.width ** 3 / 12
self.fe = BeamFE(x, 0, 0, EIy, EIz, twist=twist)
self.fe.set_boundary_conditions('C', 'F')
def test_deflection_under_tip_load_y(self):
Q = transverse_load(self.num_elements, magnitude=1, only_tip=True)
defl, reactions = self.fe.static_deflection(Q=Q)
assert_allclose(defl[-6:][1], 0.001754, atol=1e-5)
def test_deflection_under_tip_load_z(self):
Q = transverse_load(self.num_elements, magnitude=1,
angle=(np.pi / 2), only_tip=True)
defl, reactions = self.fe.static_deflection(Q=Q)
assert_allclose(defl[-6:][2], 0.005424, atol=1e-5)
def setup(self):
x = np.linspace(0, self.length, self.num_elements)
twist = x * self.tip_twist / self.length
EIy = self.E * self.width * self.depth**3 / 12
EIz = self.E * self.depth * self.width**3 / 12
self.fe = BeamFE(x, 0, 0, EIy, EIz, twist=twist)
self.fe.set_boundary_conditions('C', 'F')
def setUp(self):
x = linspace(0, 1, 16)
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0)
self.fe.set_boundary_conditions('C', 'F')
self.fe.set_dofs([False, False, True, False, True, False])
def setUp(self):
data = np.loadtxt("tests/example_blade_data.txt")
x, density, EA, EI_flap, EI_edge, twist = data.T
# Twist has been saved in my convention (+ve X rotation), not Bladed
self.fe = BeamFE(x, density, EA, EI_flap, EI_edge, twist=twist)
self.fe.set_boundary_conditions('C', 'F')
self.modal = ModalBeamFE(self.fe)
self.answers = np.load("tests/example_blade_mode_results.npz")
def setUp(self):
x = array([0.0, 10.0, 22.0, 28.0])
EI = array([[2e7, 2e7], [1e7, 1e7], [1e7, 1e7]])
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0)
self.fe.set_boundary_conditions('C', 'C')
self.fe.set_dofs([False, False, True, False, True, False])
def setup(self):
x = np.linspace(0, self.length, 10)
self.fe = BeamFE(x,
self.density,
0,
self.EIy,
self.EIz,
twist=self.twist)
self.fe.set_boundary_conditions('C', 'F')
def test_uniform_tension_increases_stiffness(self):
x = np.linspace(0, self.length, 10)
fe1 = BeamFE(x, self.density, 0, self.EI, self.EI)
fe2 = BeamFE(x,
self.density,
0,
self.EI,
self.EI,
axial_force=20 * np.ones(10))
assert np.all(np.diag(fe2.Ks) >= np.diag(fe1.Ks))
def blade_fe(blade, root_length=0.0, spin=0.0):
# positive blade twist is geometrically a negative x-rotation
twist = -np.array(blade['twist'])
x = blade['radii']
qn = interleave(root_length + np.array(x), 6)
N = BeamFE.centrifugal_force_distribution(qn, blade['density'])
fe = BeamFE(x, blade['density'], blade['EA'],
blade['EIyy'], blade['EIzz'], twist=twist,
axial_force=N * spin**2)
fe.set_boundary_conditions('C', 'F')
return fe
class UniformCantileverWithConstantTwist_Test:
length = 3.2
density = 54.3
EIy = 494.2
EIz = 654.2
twist = np.radians(76.4)
def setup(self):
x = np.linspace(0, self.length, 10)
self.fe = BeamFE(x,
self.density,
0,
self.EIy,
self.EIz,
twist=self.twist)
self.fe.set_boundary_conditions('C', 'F')
def test_mass(self):
mass = self.length * self.density
assert_allclose(self.fe.mass, mass, atol=0.5)
def test_moment_of_mass(self):
I = self.density * self.length**2 / 2
m = np.dot(self.fe.S1, self.fe.q0)
assert_allclose(m[0], I, atol=0.5)
def test_moment_of_inertia(self):
Iyy = self.density * self.length**3 / 3
J = np.einsum('p, ijpq, q -> ij', self.fe.q0, self.fe.S2, self.fe.q0)
assert_allclose(J[0, 0], Iyy, atol=0.5)
assert_allclose(J[1:, 1:], 0)
def test_deflection_under_uniform_load(self):
w = 34.2 # N/m
Q = self.fe.distribute_load(transverse_load(10, w))
defl, reactions = self.fe.static_deflection(Q)
print(reactions)
x, y, z = defl[0::6], defl[1::6], defl[2::6]
# Resolve into local blade coordinates
tw = self.twist
wy = +w * np.cos(tw)
wz = -w * np.sin(tw)
local_y = wy * self.length**4 / (8 * self.EIz) # NB y defl -> EIzz
local_z = wz * self.length**4 / (8 * self.EIy)
assert_allclose(x, 0)
assert_allclose(y[-1], local_y * np.cos(tw) - local_z * np.sin(tw))
assert_allclose(z[-1], local_z * np.cos(tw) + local_y * np.sin(tw))
assert_allclose(reactions[1], -w * self.length)
class TestBeamFE_Example62(unittest.TestCase):
def setUp(self):
x = linspace(0, 1, 16)
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0)
self.fe.set_boundary_conditions('C', 'F')
self.fe.set_dofs([False, False, True, False, True, False])
def test_mode_frequencies(self):
modal = self.fe.modal_matrices()
assert_allclose(modal.w[:4], [3.5160, 22.0345, 61.6972, 120.9019],
atol=1e-1)
class TestBeamFEAttachmentModes(unittest.TestCase):
def setUp(self):
x = array([0.0, 10.0, 22.0, 28.0])
EI = array([[2e7, 2e7], [1e7, 1e7], [1e7, 1e7]])
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0)
self.fe.set_boundary_conditions('C', 'C')
self.fe.set_dofs([False, False, True, False, True, False])
def test_simple_model(self):
am = self.fe.attachment_modes()
self.assertEqual(am.shape, (4 * 6, 2 * 2))
def setUp(self):
x = linspace(0, self.L, 15)
fe = BeamFE(x, density=self.m, EA=0, EIy=self.EI, EIz=0)
fe.set_boundary_conditions('C', 'C')
fe.set_dofs([False, False, True, False, True, False])
beam = DistalModalElementFromFE('beam', fe, num_modes=1,
damping=self.damping_coeff)
system = System()
system.add_leaf(beam)
system.setup()
system.update_kinematics()
self.beam, self.system = beam, system
class ExampleBlade_Test(unittest.TestCase):
def setUp(self):
data = np.loadtxt("tests/example_blade_data.txt")
x, density, EA, EI_flap, EI_edge, twist = data.T
# Twist has been saved in my convention (+ve X rotation), not Bladed
self.fe = BeamFE(x, density, EA, EI_flap, EI_edge, twist=twist)
self.fe.set_boundary_conditions('C', 'F')
self.modal = ModalBeamFE(self.fe)
self.answers = np.load("tests/example_blade_mode_results.npz")
def test_mass(self):
assert_allclose(self.fe.mass, 6547, atol=0.5)
def test_moment_of_mass(self):
m = np.dot(self.fe.S1, self.fe.q0)
assert_allclose(m[0], 84219, atol=0.5)
def test_moment_of_inertia(self):
J = np.einsum('p, ijpq, q -> ij', self.fe.q0, self.fe.S2, self.fe.q0)
assert_allclose(J[0, 0], 1784881, atol=0.5)
def test_first_four_modal_frequencies_match_saved_values(self):
assert_allclose(self.modal.w[:4], self.answers['freqs'], atol=1e-3)
def test_first_four_mode_shapes_match_saved_values(self):
v = self.modal.shapes
answer = self.answers['shapes']
for i in range(4):
assert_allclose_ignoring_sign(v[:, i], answer[:, i], atol=1e-5)
def test_mode_shapes_are_normalised(self):
w = self.modal.w
v = self.modal.shapes
M = self.fe.M
K = self.fe.K
for i in range(v.shape[1]):
assert_allclose(dot(v[:, i].T, dot(M, v[:, i])), 1.0)
assert_allclose(dot(v[:, i].T, dot(K, v[:, i])), w[i]**2)
def test_mode_are_sorted(self):
w = list(self.modal.w)
self.assertEqual(w, sorted(w))
def test_correct_number_of_modes_are_returned(self):
w_all, v_all = self.fe.normal_modes()
w, v = self.fe.normal_modes(4)
assert_allclose(w, w_all[:4])
for i in range(4):
assert_allclose_ignoring_sign(v[:, i], v_all[:, i])
class TestBeamFE_Example62(unittest.TestCase):
def setUp(self):
x = linspace(0, 1, 16)
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0)
self.fe.set_boundary_conditions('C', 'F')
self.fe.set_dofs([False, False, True, False, True, False])
def test_mode_frequencies(self):
modal = self.fe.modal_matrices()
assert_allclose(modal.w[:4],
[3.5160, 22.0345, 61.6972, 120.9019],
atol=1e-1)
def blade_fe(blade, root_length=0.0, spin=0.0):
# positive blade twist is geometrically a negative x-rotation
twist = -np.array(blade['twist'])
x = blade['radii']
qn = interleave(root_length + np.array(x), 6)
N = BeamFE.centrifugal_force_distribution(qn, blade['density'])
fe = BeamFE(x,
blade['density'],
blade['EA'],
blade['EIyy'],
blade['EIzz'],
twist=twist,
axial_force=N * spin**2)
fe.set_boundary_conditions('C', 'F')
return fe
class TestBeamFEAttachmentModes(unittest.TestCase):
def setUp(self):
x = array([0.0, 10.0, 22.0, 28.0])
EI = array([[2e7, 2e7],
[1e7, 1e7],
[1e7, 1e7]])
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0)
self.fe.set_boundary_conditions('C', 'C')
self.fe.set_dofs([False, False, True, False, True, False])
def test_simple_model(self):
am = self.fe.attachment_modes()
self.assertEqual(am.shape, (4*6, 2*2))
def _make_random_element(rdm):
# Choose some parameters randomly
length = rdm.uniform(1, 10)
density = rdm.uniform(0.1, 100)
EA = rdm.uniform(1e1, 1e8)
EIy = rdm.uniform(1e5, 1e8)
EIz = rdm.uniform(1e5, 1e8)
x = np.linspace(0, length, 20)
fe = BeamFE(x, density, EA, EIy, EIz)
fe.set_boundary_conditions('C', 'C')
beam = DistalModalElementFromFE('el', fe, 5)
beam._params = dict(length=length,
density=density,
EA=EA, EIy=EIy, EIz=EIz)
return beam
def setup(self):
x = np.linspace(0, self.length, self.num_elements)
twist = x * self.tip_twist / self.length
EIy = self.E * self.width * self.depth ** 3 / 12
EIz = self.E * self.depth * self.width ** 3 / 12
self.fe = BeamFE(x, 0, 0, EIy, EIz, twist=twist)
self.fe.set_boundary_conditions('C', 'F')
def test_first_mode_frequency(self):
# From Reddy1993, p. 160
x = np.linspace(0, 1, 16)
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe)
Mmodal = element.mass_ee
Kmodal = element.K
w = np.sqrt(np.diag(Kmodal) / np.diag(Mmodal))
assert_aae(w[0], 3.5160, decimal=4)
def setUp(self):
# FE model for beam - no modes, i.e. rigid
self.x = x = linspace(0, self.length, 20)
fe = BeamFE(x, density=2, EA=0, EIy=0, EIz=0)
# Build the elements
self.shaft = Hinge('shaft', [1, 0, 0])
self.roots = []
self.blades = []
self.pitch_bearings = []
for ib in range(1):
R = rotations(('x', ib * 2 * pi / 3), ('y', -pi / 2))
root_offset = dot(R, [self.root_length, 0, 0])
root = RigidConnection('root%d' % (ib + 1), root_offset, R)
bearing = Hinge('pitch%d' % (ib + 1), [1, 0, 0])
blade = ModalElementFromFE('blade%d' % (ib + 1), fe, 0)
self.shaft.add_leaf(root)
root.add_leaf(bearing)
bearing.add_leaf(blade)
self.roots.append(root)
self.blades.append(blade)
self.pitch_bearings.append(bearing)
# Build system
self.system = System()
self.system.add_leaf(self.shaft)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
class UniformCantileverWithConstantTwist_Test:
length = 3.2
density = 54.3
EIy = 494.2
EIz = 654.2
twist = np.radians(76.4)
def setup(self):
x = np.linspace(0, self.length, 10)
self.fe = BeamFE(x, self.density, 0, self.EIy, self.EIz,
twist=self.twist)
self.fe.set_boundary_conditions('C', 'F')
def test_mass(self):
mass = self.length * self.density
assert_allclose(self.fe.mass, mass, atol=0.5)
def test_moment_of_mass(self):
I = self.density * self.length ** 2 / 2
m = np.dot(self.fe.S1, self.fe.q0)
assert_allclose(m[0], I, atol=0.5)
def test_moment_of_inertia(self):
Iyy = self.density * self.length ** 3 / 3
J = np.einsum('p, ijpq, q -> ij', self.fe.q0, self.fe.S2, self.fe.q0)
assert_allclose(J[0, 0], Iyy, atol=0.5)
assert_allclose(J[1:, 1:], 0)
def test_deflection_under_uniform_load(self):
w = 34.2 # N/m
Q = self.fe.distribute_load(transverse_load(10, w))
defl, reactions = self.fe.static_deflection(Q)
print(reactions)
x, y, z = defl[0::6], defl[1::6], defl[2::6]
# Resolve into local blade coordinates
tw = self.twist
wy = +w * np.cos(tw)
wz = -w * np.sin(tw)
local_y = wy * self.length**4 / (8 * self.EIz) # NB y defl -> EIzz
local_z = wz * self.length**4 / (8 * self.EIy)
assert_allclose(x, 0)
assert_allclose(y[-1], local_y * np.cos(tw) - local_z * np.sin(tw))
assert_allclose(z[-1], local_z * np.cos(tw) + local_y * np.sin(tw))
assert_allclose(reactions[1], -w * self.length)
def setUp(self):
x = linspace(0, 1, 16)
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0)
self.fe.set_boundary_conditions('C', 'F')
self.fe.set_dofs([False, False, True, False, True, False])
def _make_random_element(rdm):
# Choose some parameters randomly
length = rdm.uniform(1, 10)
density = rdm.uniform(0.1, 100)
EA = rdm.uniform(1e1, 1e8)
EIy = rdm.uniform(1e5, 1e8)
EIz = rdm.uniform(1e5, 1e8)
x = np.linspace(0, length, 20)
fe = BeamFE(x, density, EA, EIy, EIz)
fe.set_boundary_conditions('C', 'C')
beam = DistalModalElementFromFE('el', fe, 5)
beam._params = dict(length=length,
density=density,
EA=EA,
EIy=EIy,
EIz=EIz)
return beam
def setUp(self):
x = array([0.0, 10.0, 22.0, 28.0])
EI = array([[2e7, 2e7],
[1e7, 1e7],
[1e7, 1e7]])
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0)
self.fe.set_boundary_conditions('P', 'C')
self.fe.set_dofs([False, False, True, False, True, False])
def test_rigid_element_reference_loading(self):
# Make an element with no modes (rigid)
fe = BeamFE(np.linspace(0, 1, 11), density=1, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe, 0)
# Distributed load, linearly interpolated
load = np.zeros((11, 3))
load[:, 2] = -100 # Uniform load in z direction
element.apply_distributed_loading(load)
assert_aae(element.applied_stress, [])
assert_aae(element.applied_forces[0:6], [0, 0, -100 * 1, 0, 50, 0])
assert_aae(element.applied_forces[6:12], 0)
# Distributed load, triangle at tip
load = np.zeros((11, 3))
load[-1, 2] = -100 # tapered load in z direction
element.applied_forces[:] = 0 # reset
element.apply_distributed_loading(load)
F = -100 * 0.1 / 2
assert_aae(element.applied_forces[0:6],
[0, 0, F, 0, -F * (0.9 + 0.1 * 2 / 3), 0])
assert_aae(element.applied_forces[6:12], 0)
class AppliedLoads_Tests:
def setup(self):
self.fe = BeamFE(linspace(0, 10, 11), density=0, EA=0, EIy=1, EIz=1)
def test_uniform_load(self):
f = array([3.4, 5.6, 7.2])
load = zeros(6 * 11)
for i in range(3):
load[i::6] = f[i]
# Force
F = dot(self.fe.F1, load)
assert_allclose(F, f * 10)
# Moment
q = self.fe.q0
I = np.vstack((
dot(q.T, (self.fe.F2[1, 2] - self.fe.F2[2, 1])),
dot(q.T, (self.fe.F2[2, 0] - self.fe.F2[0, 2])),
dot(q.T, (self.fe.F2[0, 1] - self.fe.F2[1, 0])),
))
Q = dot(I, load)
assert_allclose(Q, [0, -f[2] * 10 * 5, f[1] * 10 * 5])
def test_linear_load(self):
f = array([3.4, 5.6, 7.2])
load = zeros(6 * 11)
for i in range(3):
load[i::6] = linspace(0, f[i], 11)
# Force
F = dot(self.fe.F1, load)
assert_allclose(F, f * 10 / 2)
# Moment
q = self.fe.q0
I = np.vstack((
dot(q.T, (self.fe.F2[1, 2] - self.fe.F2[2, 1])),
dot(q.T, (self.fe.F2[2, 0] - self.fe.F2[0, 2])),
dot(q.T, (self.fe.F2[0, 1] - self.fe.F2[1, 0])),
))
Q = dot(I, load)
assert_allclose(
Q, [0, -f[2] * 10 / 2 * 10 * 2 / 3, f[1] * 10 / 2 * 10 * 2 / 3])
# Check stresses
assert_allclose(dot(self.fe.F, load), self.fe.distribute_load(load))
class TestBeamFE_Example42(unittest.TestCase):
def setUp(self):
x = array([0.0, 4.0, 10.0])
EI = 144.0
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0)
self.fe.set_boundary_conditions('C', 'F')
self.fe.set_dofs([False, False, True, False, True, False])
def test_stiffness(self):
# Reddy1993, p 164
expected = array([
[27, 0, 0, 0, 0, 0],
[-54, 144, 0, 0, 0, 0],
[-27, 54, 27 + 8, 0, 0, 0],
[-54, 72, 54 - 24, 144 + 96, 0, 0],
[0, 0, -8, 24, 8, 0],
[0, 0, -24, 48, 24, 96],
])
expected += expected.T - np.diag(expected.diagonal()) # make symmetric
expected_II = expected[2:6, 2:6]
expected_IB = expected[2:6, 0:2]
assert_allclose(self.fe.K_II, expected_II)
assert_allclose(self.fe.K_IB, expected_IB)
def test_distributed_load(self):
# Distributed force on second element:
load = array([0, 0, 0, 0, 0, 0, 0, 0, -100, 0, 0, 0])
Q = self.fe.distribute_load_on_element(1, load)
Q = Q[[2, 4, 8, 10, 14, 16]]
assert_allclose(Q, [0, 0, -90, 120, -210, -180])
def test_solution_without_spring(self):
# Distributed force on first element:
load = array([0, 0, 0, 0, 0, 0, 0, 0, -100, 0, 0, 0])
QF = self.fe.distribute_load_on_element(1, load)
# Solve static deflection
deflections, reactions = self.fe.static_deflection(QF)
# Expected values from Reddy
expected_deflections = zeros_like(deflections)
expected_deflections[[
8, 10, 14, 16
]] = array([-1.6, 0.72, -7.108, 0.99]) * 1e4 / 143.0
assert_allclose(deflections, expected_deflections, atol=1e1)
class TestBeamFE_Example42(unittest.TestCase):
def setUp(self):
x = array([0.0, 4.0, 10.0])
EI = 144.0
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0)
self.fe.set_boundary_conditions('C', 'F')
self.fe.set_dofs([False, False, True, False, True, False])
def test_stiffness(self):
# Reddy1993, p 164
expected = array([
[27, 0, 0, 0, 0, 0],
[-54, 144, 0, 0, 0, 0],
[-27, 54, 27+8, 0, 0, 0],
[-54, 72, 54-24, 144+96, 0, 0],
[0, 0, -8, 24, 8, 0],
[0, 0, -24, 48, 24, 96],
])
expected += expected.T - np.diag(expected.diagonal()) # make symmetric
expected_II = expected[2:6, 2:6]
expected_IB = expected[2:6, 0:2]
assert_allclose(self.fe.K_II, expected_II)
assert_allclose(self.fe.K_IB, expected_IB)
def test_distributed_load(self):
# Distributed force on second element:
load = array([0, 0, 0, 0, 0, 0, 0, 0, -100, 0, 0, 0])
Q = self.fe.distribute_load_on_element(1, load)
Q = Q[[2, 4, 8, 10, 14, 16]]
assert_allclose(Q, [0, 0, -90, 120, -210, -180])
def test_solution_without_spring(self):
# Distributed force on first element:
load = array([0, 0, 0, 0, 0, 0, 0, 0, -100, 0, 0, 0])
QF = self.fe.distribute_load_on_element(1, load)
# Solve static deflection
deflections, reactions = self.fe.static_deflection(QF)
# Expected values from Reddy
expected_deflections = zeros_like(deflections)
expected_deflections[[8, 10, 14, 16]] = array(
[-1.6, 0.72, -7.108, 0.99]) * 1e4 / 143.0
assert_allclose(deflections, expected_deflections, atol=1e1)
class AppliedLoads_Tests:
def setup(self):
self.fe = BeamFE(linspace(0, 10, 11), density=0, EA=0, EIy=1, EIz=1)
def test_uniform_load(self):
f = array([3.4, 5.6, 7.2])
load = zeros(6 * 11)
for i in range(3): load[i::6] = f[i]
# Force
F = dot(self.fe.F1, load)
assert_allclose(F, f * 10)
# Moment
q = self.fe.q0
I = np.vstack((
dot(q.T, (self.fe.F2[1, 2] - self.fe.F2[2, 1])),
dot(q.T, (self.fe.F2[2, 0] - self.fe.F2[0, 2])),
dot(q.T, (self.fe.F2[0, 1] - self.fe.F2[1, 0])),
))
Q = dot(I, load)
assert_allclose(Q, [0, -f[2]*10*5, f[1]*10*5])
def test_linear_load(self):
f = array([3.4, 5.6, 7.2])
load = zeros(6 * 11)
for i in range(3): load[i::6] = linspace(0, f[i], 11)
# Force
F = dot(self.fe.F1, load)
assert_allclose(F, f * 10 / 2)
# Moment
q = self.fe.q0
I = np.vstack((
dot(q.T, (self.fe.F2[1, 2] - self.fe.F2[2, 1])),
dot(q.T, (self.fe.F2[2, 0] - self.fe.F2[0, 2])),
dot(q.T, (self.fe.F2[0, 1] - self.fe.F2[1, 0])),
))
Q = dot(I, load)
assert_allclose(Q, [0, -f[2]*10/2 * 10*2/3, f[1]*10/2 * 10*2/3])
# Check stresses
assert_allclose(dot(self.fe.F, load), self.fe.distribute_load(load))
def test_static_deflection(self):
x = array([0.0, 4.0, 10.0])
EI = 144.0
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
fe = BeamFE(x, density=10, EA=0, EIy=EI, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe)
# Distributed load, linearly interpolated
load = np.zeros((3, 3))
load[-1, 2] = -100 # Load in z direction at tip
element.apply_distributed_loading(load)
defl = -element.applied_stress / np.diag(element.K)
# Check against directly calculating static deflection from FE
Q = fe.distribute_load(interleave(load, 6))
defl_fe, reactions_fe = fe.static_deflection(Q)
assert_aae(dot(element.shapes, defl), defl_fe, decimal=2)
def setUp(self):
x = linspace(0, self.L, 15)
fe = BeamFE(x, density=self.m, EA=0, EIy=self.EI, EIz=0)
fe.set_boundary_conditions('C', 'C')
fe.set_dofs([False, False, True, False, True, False])
beam = DistalModalElementFromFE('beam',
fe,
num_modes=1,
damping=self.damping_coeff)
system = System()
system.add_leaf(beam)
system.setup()
system.update_kinematics()
self.beam, self.system = beam, system
def test_first_mode_frequency(self):
# From Reddy1993, p. 160
x = np.linspace(0, 1, 16)
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe)
Mmodal = element.mass_ee
Kmodal = element.K
w = np.sqrt(np.diag(Kmodal) / np.diag(Mmodal))
assert_aae(w[0], 3.5160, decimal=4)
def test_static_deflection(self):
x = array([0.0, 4.0, 10.0])
EI = 144.0
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
fe = BeamFE(x, density=10, EA=0, EIy=EI, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe)
# Distributed load, linearly interpolated
load = np.zeros((3, 3))
load[-1, 2] = -100 # Load in z direction at tip
element.apply_distributed_loading(load)
defl = -element.applied_stress / np.diag(element.K)
# Check against directly calculating static deflection from FE
Q = fe.distribute_load(interleave(load, 6))
defl_fe, reactions_fe = fe.static_deflection(Q)
assert_aae(dot(element.shapes, defl), defl_fe, decimal=2)
def test_rigid_element_reference_loading(self):
# Make an element with no modes (rigid)
fe = BeamFE(np.linspace(0, 1, 11), density=1, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe, 0)
# Distributed load, linearly interpolated
load = np.zeros((11, 3))
load[:, 2] = -100 # Uniform load in z direction
element.apply_distributed_loading(load)
assert_aae(element.applied_stress, [])
assert_aae(element.applied_forces[0:6], [0, 0, -100 * 1, 0, 50, 0])
assert_aae(element.applied_forces[6:12], 0)
# Distributed load, triangle at tip
load = np.zeros((11, 3))
load[-1, 2] = -100 # tapered load in z direction
element.applied_forces[:] = 0 # reset
element.apply_distributed_loading(load)
F = -100 * 0.1 / 2
assert_aae(element.applied_forces[0:6],
[0, 0, F, 0, -F * (0.9 + 0.1 * 2 / 3), 0])
assert_aae(element.applied_forces[6:12], 0)
def setup(self):
self.fe = BeamFE(linspace(0, 10, 11), density=0, EA=0, EIy=1, EIz=1)
class TestBeamFE_Example41(unittest.TestCase):
def setUp(self):
x = array([0.0, 10.0, 22.0, 28.0])
EI = array([[2e7, 2e7], [1e7, 1e7], [1e7, 1e7]])
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0)
self.fe.set_boundary_conditions('P', 'C')
self.fe.set_dofs([False, False, True, False, True, False])
def test_stiffness(self):
# Reddy1993, pp 161-162
expected = 1e7 * array([
[0.024, -0.12, -0.024, -0.12, 0, 0, 0, 0],
[0, 0.80, 0.12, 0.40, 0, 0, 0, 0],
[0, 0, 0.0309, 0.0783, -0.00694, -0.04167, 0, 0],
[0, 0, 0, 1.133, 0.0417, 0.167, 0, 0],
[0, 0, 0, 0, 0.0625, -0.125, -0.0556, -0.167],
[0, 0, 0, 0, 0, 1, 0.1667, 0.333],
[0, 0, 0, 0, 0, 0, 0.0556, 0.1667],
[0, 0, 0, 0, 0, 0, 0, 0.6667],
])
expected += expected.T - np.diag(expected.diagonal()) # make symmetric
expected_II = expected[1:6, 1:6]
expected_IB = expected[1:6, [0, 6, 7]]
assert_allclose(expected_II, self.fe.K_II, atol=1e-2 * 1e7)
assert_allclose(expected_IB, self.fe.K_IB, atol=1e-3 * 1e7)
def test_distributed_load(self):
# Distributed force on first element:
load = array([0, 0, -2400, 0, 0, 0, 0, 0, -2400, 0, 0, 0])
Q = self.fe.distribute_load_on_element(0, load)
Q = Q[[2, 4, 8, 10, 14, 16, 20, 22]]
assert_allclose(Q / 1e3, [-12, 20, -12, -20, 0, 0, 0, 0])
def test_solution(self):
# Distributed force on first element:
load = array([0, 0, -2400, 0, 0, 0, 0, 0, -2400, 0, 0, 0])
QF = self.fe.distribute_load_on_element(0, load)
# Nodal forces:
Q = np.zeros(self.fe.K.shape[0])
Q[14] = -10000 # z force at 3rd node
# Solve static deflection
deflections, reactions = self.fe.static_deflection(QF + Q)
# Expected values from Reddy
expected_deflections = zeros_like(deflections)
expected_deflections[[4, 8, 10, 14, 16]] = [
0.03856, -0.2808, 0.01214, -0.1103, -0.02752
]
assert_allclose(deflections, expected_deflections, atol=1e-4)
expected_reactions = zeros_like(reactions)
expected_reactions[[2, 20, 22]] = [18565.54, 15434.46, 92164.83]
assert_allclose(reactions, expected_reactions, atol=1e-2)
def test_load_matrix_gives_same_result_as_method(self):
# Distributed force on all elements
load = np.zeros(4 * 6)
load[2::6] = [350, 324, 654, 54] # Z component
Q1 = self.fe.distribute_load(load)
Q2 = dot(self.fe.F, load)
assert_allclose(Q1, Q2)
class UniformCantilever_Test:
length = 3.2
density = 54.3
EI = 494.2
def setup(self):
x = np.linspace(0, self.length, 10)
self.fe = BeamFE(x, self.density, 0, self.EI, self.EI)
self.fe.set_boundary_conditions('C', 'F')
def test_mass(self):
mass = self.length * self.density
assert_allclose(self.fe.mass, mass, atol=0.5)
def test_moment_of_mass(self):
I = self.density * self.length**2 / 2
m = np.dot(self.fe.S1, self.fe.q0)
assert_allclose(m[0], I, atol=0.5)
def test_moment_of_inertia(self):
Iyy = self.density * self.length**3 / 3
J = np.einsum('p, ijpq, q -> ij', self.fe.q0, self.fe.S2, self.fe.q0)
assert_allclose(J[0, 0], Iyy, atol=0.5)
assert_allclose(J[1:, 1:], 0)
def test_deflection_with_tip_load(self):
# Simple tip load should produce databook tip deflection
W = 54.1 # N
Q = transverse_load(10, W, only_tip=True)
defl, reactions = self.fe.static_deflection(Q=Q)
x, y, z = defl[0::6], defl[1::6], defl[2::6]
assert_allclose(x, 0)
assert_allclose(z, 0)
assert_allclose(y[-1], W * self.length**3 / (3 * self.EI))
def test_deflection_under_uniform_load(self):
# Simple uniform load should produce databook tip deflection
w = 34.2 # N/m
Q = self.fe.distribute_load(transverse_load(10, magnitude=w))
defl, reactions = self.fe.static_deflection(Q)
x, y, z = defl[0::6], defl[1::6], defl[2::6]
assert_allclose(x, 0)
assert_allclose(z, 0)
assert_allclose(y[-1], w * self.length**4 / (8 * self.EI))
def test_deflection_is_parallel_to_uniform_loading(self):
w = 67.4 # distributed load (N/m)
theta = np.radians(35.4) # angle from y axis of load
Q = self.fe.distribute_load(transverse_load(10, w, theta))
defl, reactions = self.fe.static_deflection(Q)
x, y, z = defl[0::6], defl[1::6], defl[2::6]
assert_allclose(x, 0)
load_angle = np.arctan2(z, y)
assert_allclose(load_angle[1:], theta)
def test_reaction_force(self):
# Simple uniform load should produce databook tip deflection
w = 34.2 # N/m
F = transverse_load(10, w)
# Reaction force is F1 * F
R = np.dot(self.fe.F1, F)
assert_allclose(R, [0, w * self.length, 0])
class UniformCantilever_Test:
length = 3.2
density = 54.3
EI = 494.2
def setup(self):
x = np.linspace(0, self.length, 10)
self.fe = BeamFE(x, self.density, 0, self.EI, self.EI)
self.fe.set_boundary_conditions('C', 'F')
def test_mass(self):
mass = self.length * self.density
assert_allclose(self.fe.mass, mass, atol=0.5)
def test_moment_of_mass(self):
I = self.density * self.length ** 2 / 2
m = np.dot(self.fe.S1, self.fe.q0)
assert_allclose(m[0], I, atol=0.5)
def test_moment_of_inertia(self):
Iyy = self.density * self.length ** 3 / 3
J = np.einsum('p, ijpq, q -> ij', self.fe.q0, self.fe.S2, self.fe.q0)
assert_allclose(J[0, 0], Iyy, atol=0.5)
assert_allclose(J[1:, 1:], 0)
def test_deflection_with_tip_load(self):
# Simple tip load should produce databook tip deflection
W = 54.1 # N
Q = transverse_load(10, W, only_tip=True)
defl, reactions = self.fe.static_deflection(Q=Q)
x, y, z = defl[0::6], defl[1::6], defl[2::6]
assert_allclose(x, 0)
assert_allclose(z, 0)
assert_allclose(y[-1], W * self.length**3 / (3 * self.EI))
def test_deflection_under_uniform_load(self):
# Simple uniform load should produce databook tip deflection
w = 34.2 # N/m
Q = self.fe.distribute_load(transverse_load(10, magnitude=w))
defl, reactions = self.fe.static_deflection(Q)
x, y, z = defl[0::6], defl[1::6], defl[2::6]
assert_allclose(x, 0)
assert_allclose(z, 0)
assert_allclose(y[-1], w * self.length**4 / (8 * self.EI))
def test_deflection_is_parallel_to_uniform_loading(self):
w = 67.4 # distributed load (N/m)
theta = np.radians(35.4) # angle from y axis of load
Q = self.fe.distribute_load(transverse_load(10, w, theta))
defl, reactions = self.fe.static_deflection(Q)
x, y, z = defl[0::6], defl[1::6], defl[2::6]
assert_allclose(x, 0)
load_angle = np.arctan2(z, y)
assert_allclose(load_angle[1:], theta)
def test_reaction_force(self):
# Simple uniform load should produce databook tip deflection
w = 34.2 # N/m
F = transverse_load(10, w)
# Reaction force is F1 * F
R = np.dot(self.fe.F1, F)
assert_allclose(R, [0, w * self.length, 0])
def setup(self):
x = np.linspace(0, self.length, 10)
self.fe = BeamFE(x, self.density, 0, self.EIy, self.EIz,
twist=self.twist)
self.fe.set_boundary_conditions('C', 'F')
def setup(self):
self.fe = BeamFE(linspace(0, 10, 11), density=0, EA=0, EIy=1, EIz=1)
class TestBeamFE_Example41(unittest.TestCase):
def setUp(self):
x = array([0.0, 10.0, 22.0, 28.0])
EI = array([[2e7, 2e7],
[1e7, 1e7],
[1e7, 1e7]])
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
self.fe = BeamFE(x, density=0, EA=0, EIy=EI, EIz=0)
self.fe.set_boundary_conditions('P', 'C')
self.fe.set_dofs([False, False, True, False, True, False])
def test_stiffness(self):
# Reddy1993, pp 161-162
expected = 1e7 * array([
[0.024, -0.12, -0.024, -0.12, 0, 0, 0, 0],
[0, 0.80, 0.12, 0.40, 0, 0, 0, 0],
[0, 0, 0.0309, 0.0783, -0.00694, -0.04167, 0, 0],
[0, 0, 0, 1.133, 0.0417, 0.167, 0, 0],
[0, 0, 0, 0, 0.0625, -0.125, -0.0556, -0.167],
[0, 0, 0, 0, 0, 1, 0.1667, 0.333],
[0, 0, 0, 0, 0, 0, 0.0556, 0.1667],
[0, 0, 0, 0, 0, 0, 0, 0.6667],
])
expected += expected.T - np.diag(expected.diagonal()) # make symmetric
expected_II = expected[1:6, 1:6]
expected_IB = expected[1:6, [0, 6, 7]]
assert_allclose(expected_II, self.fe.K_II, atol=1e-2 * 1e7)
assert_allclose(expected_IB, self.fe.K_IB, atol=1e-3 * 1e7)
def test_distributed_load(self):
# Distributed force on first element:
load = array([0, 0, -2400, 0, 0, 0, 0, 0, -2400, 0, 0, 0])
Q = self.fe.distribute_load_on_element(0, load)
Q = Q[[2, 4, 8, 10, 14, 16, 20, 22]]
assert_allclose(Q / 1e3, [-12, 20, -12, -20, 0, 0, 0, 0])
def test_solution(self):
# Distributed force on first element:
load = array([0, 0, -2400, 0, 0, 0, 0, 0, -2400, 0, 0, 0])
QF = self.fe.distribute_load_on_element(0, load)
# Nodal forces:
Q = np.zeros(self.fe.K.shape[0])
Q[14] = -10000 # z force at 3rd node
# Solve static deflection
deflections, reactions = self.fe.static_deflection(QF + Q)
# Expected values from Reddy
expected_deflections = zeros_like(deflections)
expected_deflections[[4, 8, 10, 14, 16]] = [0.03856, -0.2808, 0.01214,
-0.1103, -0.02752]
assert_allclose(deflections, expected_deflections, atol=1e-4)
expected_reactions = zeros_like(reactions)
expected_reactions[[2, 20, 22]] = [18565.54, 15434.46, 92164.83]
assert_allclose(reactions, expected_reactions, atol=1e-2)
def test_load_matrix_gives_same_result_as_method(self):
# Distributed force on all elements
load = np.zeros(4 * 6)
load[2::6] = [350, 324, 654, 54] # Z component
Q1 = self.fe.distribute_load(load)
Q2 = dot(self.fe.F, load)
assert_allclose(Q1, Q2)
