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))
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 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 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 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)
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_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)
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))
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)
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_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 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 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 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)